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 Invisible Mathematics in Italo Calvino’s Le città invisibili Ileana Moreno-­‐Viqueira Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2013 © 2013
Ileana Moreno-Viqueira
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ABSTRACT
Invisible Mathematics in Italo Calvino’s Le città invisibili
Ileana Moreno-Viqueira
This dissertation examines the use of mathematical concepts as an essential structural and
thematic element in Italo Calvino’s Le città invisibili. The author‘s conception of literature as a
combinatorial art, intrinsically mathematical itself, is the point of departure. Focal to the study is
Calvino’s interest in that which is an essential part of the combinatorial game and the key to
Gödel Incompleteness Theory, namely, the elements of surprise and the unexpected - the
exceptions to the rule.
Other critical approaches to Calvino’s work, like semiotic, structuralism and scientific are
interrelated to Mathematics, but what this study proposes is a strictly mathematical approach to
complement that which has already been pointed out. A mathematical perspective based on an
understanding of Mathematics as more than just numbers encompasses the whole analysis.
Mathematics is given its proper place as a humanistic discipline. It is an interdisciplinary
proposal of literature and science, pertinent to Calvino’s writing.
The purpose is to unveil a “hidden math” which from the perspective of this study is an
intrinsic tool in Calvino’s writing process of Città. As a versatile writer, Calvino manages to use
mathematics in such subtle ways that it may not be perceptible at first sight. Most importantly,
within these mathematical concepts and images lies, in part, the potential character of literature
for which the author aims: that latent yet invisible possibility, that search for new forms (like the
cities). These ideas, particularly related to potential literature, are also analyzed from his interest
and involvement in Oulipo (Ouvroir de Littérature Potentielle).
The study begins by unfolding what aspects of combinatorial mathematics are present in
Le città invisibili; how these concepts as well as other images are used in the construction and
design of the cities and the book; and to find out why Calvino finds recourse to mathematics as a
narrative and creative strategy.
Calvino’s use of mathematical concepts are studied as a “visual instrument” in the
organization and construction of his imaginative writing and, furthermore, as a means to achieve
“lightness” structurally and thematically through the abstract, aesthetic and, at times, even
humorous nature of mathematics. In their own way mathematics and literature attempt to make
visible what is invisible, and they both struggle to remove weight from their own “systems” of
expression.
In conclusion, the investigation intends to demonstrate through Calvino’s Le città
invisibili, how mathematics and literature complement each other in the search for new forms,
new ideas, new stories.
Table of Contents
Introduction
1
Chapter One: Genesis of the Book
1.1 Story of a design
14
1.2 From Description to Structure
21
1.3 Design of a Story: The Index
26
1.4 Design of a Story: the Frame
33
Chapter Two: The Book as Space
2.1 Invisible spaces
50
2.1.1 Space as a Combinatorial Construction
54
2.1.2 Combinatorial Game and the Unexpected
62
2.1.3 The Book and the City
71
2. 2 Traveling Through the Cities
80
2.2.1 Mathematical Constructions
82
2.2.2 Doubling doubles
109
Chapter Three: Space in Search of a Form
121
3.1 Motion in Space: The Game of Chess
124
3.2 Spaces in Motion: Complex Models
136
3.3 The Center: Lightness and the Spiral
146
Conclusion
184
Bibliography
190
1
Introduction
Se puede comprobar fácilmente que Newton y Kepler modelaron sus
órbitas planetarias de una manera esencialmente geométrica. Sin embargo,
las elipsis en sí no tienen existencia física en el espacio, son tan sólo
senderos invisibles trazados por planetas en órbita. (Richard Mankiewickz.
Historia de las matemáticas. Del cálculo al caos.100)1
Not often are literature and mathematics perceived as being intricately linked
together, however, through the ages this link between mathematics, poetry and literature
is evident. This is precisely the case in Italo Calvino’s Le città invisibili.2 Throughout
his writing career, Italo Calvino left many important views regarding the essence of
literature. Among these, the definition of “literature as a combinatorial game” stands out.
From this perspective of the concept of literature as a “combinatorial game” I intend to
focus my investigation of Calvino’s use of mathematical concepts in Le città invisibili. 3
The present study will examine how in this particular book mathematical concepts
constitute an essential element both structurally as well as thematically.4 Although some
attention has been paid to the more explicit aspects of combinatorial structures of his texts (particularly, in Il castello dei destini incrociati) further complexities and subtleties of his combinatorial mathematics have been, if not completely neglected, at
least not fully perceived. Among his many writings, Le città invisibili, the book which the
author claims that he “managed to say the most”, presents itself as ideal to pursue the
1
“One could easily prove that Newton and Kepler modeled their planetary orbits in an
2
“Symmetries and arithmetic have always tempted Calvino’s imagination to grow flirtatious and to
begin its fantastic displays” (Seamus Heaney 77).
3
Another contemporary critical study regarding mathematics and literature is Guillermo Martínez “Borges y
la matemática” (2012).
2
hypothesis that his use of mathematical concepts, such as combinatorial art, goes beyond
what has been analyzed so far.
Calvino’s work has been the object of a vast amount of criticism (McLaughlin
1998) from a wide range of approaches. The main focus has been in aspects other than
mathematical, which is fundamental to fields of study such as science; also linguistics,
structuralism, semiotics, post-modernism. The critical research from these perspectives
shows links to their mathematical basis. Studies referring to the scientific interests of the
author have placed much emphasis, for instance, in Cosmicomics, but also in his essays
and their repercussions in other writings. Massimo Bucciantini in Italo Calvino e la
scienza (2007) analyzes Calvino’s relationship with science as well as his scientific
images and language. Concerning mathematics, Gabrielle Lolli’s Discorso sulla
matematica (2011) proposes reading Italo Calvino’s Six Memos for the Next Millennium
“come una parabola sulla matematica”. Furthermore, Calvino’s books have been used
with a didactic purpose to illustrate mathematics, particularly, Città. Recently, the visual
feature of his work has had a significant impact on critical studies (Ricci 2001, Belpoliti
2006, Ragusa 1983). The theme of the city –a constant topic throughout most of Calvino’s literary career – has been the object of critical works with an architectural
approach, for example, Modena (2011). All in all, recent criticism keeps pointing to
Calvino’s interdisciplinary view on literature, while the overall structure of Città, as
designed by Calvino has been commented by the author and the critic.
This study intends to complement the research accomplished at present,
particularly on Calvino’s visual, scientific, architectural and other interdisciplinary
interests. Mathematical concepts are at the basis of these approaches, yet they are merely
3
mentioned and, generally, in other contexts. The purpose of this dissertation is to
demonstrate a mathematical perspective, which has been pointed out, but not particularly
studied in extension or depth. From my perspective, this “hidden math” is an intrinsic
tool in his creative process of writing. As a versatile writer, he managed to use
mathematics in such subtle ways that may not be perceptible at first sight. Most
importantly, within these concepts and images lies, in part, the “potential” character of
literature that the author was aiming for.
The term ars combinatoria was inspired by Leibniz and his famous Dissertatio de
Arte Combinatoria (1666), following Giordano Bruno’s admiration of the Medieval
Catalan monk Ramón Lull ‘Ars Magna (1305). In essence, this ‘art’ consists in nothing
but arranging a limited number of elements subject to certain rules. By altering the
sequences, different arrangements are formed. The goal becomes to create as many
combinations as possible.
For Calvino, narrative is a combinatorial game “[which] plays with possibilities
intrinsic to its own material”. This ars combinatoria, he adds, is “a game that at a certain
point is invested with an unexpected meaning.” (“Cybernetics and Ghosts” 22;
“Cibernetica e fantasmi” 210)
Combinatorial systems are by definition formal systems. Thus, according to
Gödel’s Incompleteness Theorem, they cannot escape a fundamental paradox.
Unavoidably, within any system, something must escape, something that bypassing its
own rules becomes, in every sense, surprisingly exceptional: “In every system, something
emerges which is not truly part of the system” (D. Hofstadter 3).
4
Even if these statements may seem, at first, potentially contradictory, they are, as
will be here examined, complementary. In fact, this paradoxical character of the author’s
writing will be essential in the present discussion. Precisely this controversial element of
surprise, the exceptions to the rule, will be the argumentative focus of this research. The
goal is to search, in order to discover exactly the manner in which Calvino employs
combinatorial mathematics, not just to construct his book but also to involve the reader in
the design of his stories, by engaging him in the game of reading.
Calvino’s work displays a constant search for new forms. Each of his works has
been contemplated as another coordinate in this path to innovation (Warren Motte 1986).
Refusing to see the combinatorial game as verbal arithmetic, Calvino insists on the
liberating ‘potential’ of ars combinatoria. Literature, he convincingly argues, has always
been “a struggle to escape the confines of language”, to break barriers, to search for new
forms, to say what has not been said. Because the combinatorial process can
spontaneously transcend itself, exceptional storytelling occurs.
Similarly, mathematics constantly struggles to search, for new designs, new ways
to look at things, to break the limits already established. Mathematical concepts become
potential tools capable not just to enumerate, catalogue, organize or map the possible
combinations but also to be used as a creative narrative method, as a stimulus for
creativity and fantasy.
Mathematics does not involve “just numbers”, but concepts, many times shared
with other disciplines: duplicity, reversibility, maps, networks and geometric figures are
only some examples. They are used to see what is not obvious, such as patterns, or even
disorder; to demonstrate new ways to look at things as well as new things to look at, in
5
order to understand or mentally visualize; simplify complexities. Mostly, they constitute
essential strategies for mental agility. Consequently, they are useful for the creative
process of the mind, be it scientific, artistic or literary.
Doubles, spirals, maps, puzzles, labyrinths, chessboards, numbers, geometrical
figures, abound in his texts, as if through these images, the author were trying to draw, to
paint, to design imaginary spaces for our minds to inhabit. But most importantly, the
potentiality of the combinatorial structure, as will be demonstrated, is essentially due to
its mathematical nature.
Critics have pointed Calvino’s years in Paris (from the1960’s to 1980) as a major
turning point in his writing career. During the 60’s Calvino moved to Paris, where he
became involved with the group called Ou.Li.Po, becoming a formal member from 1973
to 1980. Founded by Raymond Queneau in 1960, - Oulipo, short for Ouvroir de
Littérature Potentielle (Workshop of Potential Literature) - was formed by a Parisian
group of mathematicians and writers. While experimenting with the possibility of
incorporating mathematical structures in the process of literary creation, they came to the
conclusion that restrictions or constraints paradoxically increased the potential of literary
invention.
According to their First Manifesto, “Mathematics – particularly the abstract
structures of contemporary mathematics- proposes thousands of possibilities for
exploration, both algebraically (recourse to new laws of composition) and topologically
(considerations of textual contiguity, openness and closure)” (Francois Le Lyonnais.
Oulipo Laboratory, ix-xx, 1995).
6
Calvino’s own expressions as to the importance of his association to this group
points out to their mathematical influence. Not surprisingly, this is precisely the period in
which Le città invisibili was conceived. Among his various “oulipian” works are “Prose
et anti combinatoire”, ”Piccolo silabario illustrato” and his translation of Queneau’s Les
Fleurs bleus.
For Franco Ricci it is no coincidence that this was the period when Calvino’s
writings were more visual (106). Besides, the fact that Calvino considered this book to be
“molto oulipiano sopra tutto per il suo indice” provides another essential point of
departure for this research.
Methodology, Structure and Chapter Development
The purpose of my research is: (1) to describe what aspects of combinatorial
mathematics appear in Italo Calvino’s Le città invisibili, (2) to see how he employs these
concepts and images on the construction and design of his cities and his book, and (3) to
(try to) find out why does he have recourse to to mathematics as a both a narrative tool
and a strategy for creativity. Furthermore, my intention is not just to investigate how
Calvino utilizes mathematical concepts as a “visual instrument” to organize and construct
to his imaginative writing, but as a means to achieve “lightness” both structurally and
thematically through the abstract, aesthetic and, at times, even humorous nature of
mathematics. Ironically, in their own way both mathematics and literature attempt to
make visual what is invisible, and they both struggle to subtract weight from their own
‘systems’ of expression. Besides, the concept of game playing as a mathematical strategy
is well integrated in the book, and needs to be part of the overall analysis.
7
The analysis will depend primarily on Calvino’s own critical works, especially,
his last legacy, Lezioni americane (1985), (Six Memos for the Next Millennium, 1993),
will be essential to this research. Precisely, in this book, in his essay “Exactitude”
Calvino explains why his Città was “the book in which he managed to say the most”.
According to him, “the complex symbol of the city gave him the greatest possibility of
expressing the tension between rational geometry and the entanglements of human
existence”. The city, he explains, “allowed him to concentrate on a single symbol all his
reflections, experiences and conjectures but to “construct a multifaceted structure, a
network in which one can follow multiple routes and follow multiple ramified
conclusions” (Six Memos for the Next Millennium 88-89).
These two aspects, the city symbol and the structure, as a net, which allowed for
multiple courses and interpretations, reinforce the importance of the architectural aspect
of the book. This “construction”, as we will see, is, in essence, mathematical. He did not
just construct a book of cities. They are more than a collection, or an addition. By
integration, Calvino’s work could be contemplated as a book-city or city- book; like a
modern city, which can be entered, traversed and exited from in multiple ways.
Through the image of the city, its variations and combinations, Calvino allows his
readers to ‘see’ the designing’ of the book as if it were a work of architecture. He creates
an analogy between the construction of the book and that of the imaginary cities. The
reading of this city-book or book-city, transforms a textual space into a “mental space”
and vice-versa. Travelling through the spaces designed by these invisible cities becomes
travelling through the text. In fact, Calvino describes a book as a space: ”uno spazio in
cui il lettore deve entrare, girare magari perdersi”. However, the author insists a book
8
must also have “una costruzione, cioè si deve poter scoprire un intreccio, un itinerario,
una soluzione” (Le città invisibili vi).
To begin, on the first chapter I will concentrate specifically on the genesis of the
book; that is, its construction and organization. According to the author, the book was
constructed fragmentarily: all of the 55 cities were created first; only after completing
this part, did he organize them through the index and the frame. He seems to have
devoted about the same amount of effort to both the creation of the cities and to their
subsequent organization, particularly the index. “Di qui l’importanza cruciale della
costruzione dell’indice, alla quale Calvino si è dedicato – ne siamo persuasi – con un
impegno pari a quello profuso nella stesura dei testi” (Mario Barenghi, Italo Calvino, le
linee e i margini 269).
Regarding this fragmentary genesis, the basic elements need to be identified.
Also, the techniques employed by the author are fundamental. His ideas on rapidity and
exactitude will be essential points of reference for the analysis of this part of the
construction. Secondly, I will analyse the combinatorial organization he gave to these
pieces, as established by the index (which leads to multiplicity) and the functions of the
frame. Beyond containing the stories, the analysis will examine from a mathematical
perspective how does the frame achieve to link the stories as it comments on them,
whether there are any other functions to the frame and how are these achieved, and what
is the purpose of framing these city spaces.
For the analysis of the index, (which originally was strategically located at the
beginning of the book by the author), we need to look at how the book became a book,
9
how it was designed. The fact that Calvino explicitly considered this book in particular
to be “molto oulipiano sopra tutto per il suo indice” gives us a starting point.
The second chapter “The Book as Space”, perhaps the most substantial part to my
research, consists on how mathematics are employed in this space, this book- city or citybook. The book is analyzed as a space through the active involvement in the game of
reading as a creative process, as a challenge proposed by the text.
The chapter includes two parts: 1) “Invisible spaces”, 2) “Traveling through the
Cities”. The first part (“Invisible Spaces”) is subdivided in three study themes: 1.1)
“Space as a Combinatorial Construction”; 1.2) “Combinatorial Game and the
Unexpected”; 1.3)”The Book and the City”. In the first theme (“Space as a Combinatorial
Construction”), the fictional space suggests a combinatorial construction, which
essentially correlates the concepts of mapping and of mathematical game playing. The
second theme, (“Combinatorial Game and the Unexpected”), points to the essential
aspects of this creative process, that of the unexpected, that element of surprise which
continues to escape the system, defying to be deciphered. The third theme or subpart,
(”The Book and the City”), examines some of the multiple instances of the theme of the
city in Calvino’s writings, in order to arrive at the correspondence created in his Invisible
Cities, an analogy, or mathematically, a mapping, an isomorphism in which one points to
the other, combining into one.
The second part of the chapter, “Traveling through the Cities”, is subdivided in
two themes: 2.1) “Mathematical constructions, 2.2) “Doubling Doubles”. In the first
section, the relevance of mathematical concepts in the construction of the cities will be
examined pointing out a mathematical thread from point to line, geometric figures, etc. In
10
the next section, “Doubling Doubles”, the study will follow the mathematical thread from
binary relationships and duplicity to multiplicity, networks, reversals, inversions, over
lapping, lack of limits, which lead to further complexities… spaces looking for a form.
The visual aspect will be essential in the analysis of the use of mathematical
concept within the cities. Duplicity, and accordingly, reversibility, ironic twists are
essential aspects of his writing. In regards to this book in particular, Calvino states
”Nelle Cttà invisibili, ogni concetto e ogni valore si rivela duplice: anche l’essattezza.”
Besides duplicity and its related concepts of reversals and multiplicity, many
mathematical images appear in this small “spaces” of the book. Geometric figures, such
as concentric circles, spheres, and spirals abound. Just as in Città, the challenge of
illustrating something invisible is central to mathematics.5
The third chapter, “Space in search of a form” will be focused on examining how
the mathematical concepts involved in the combinatorial game manage to design
different forms – regardless of their complexities – within this book space. These
combinatorial potential forms – mostly based on mobility and lightness – attempt to
manage further intricacies involving continuity and evasiveness (the spiral); multiplicity
and unity, through ephemeral and threadlike light structures (the net). Visualizing the
invisible cities comprises recognizing and understanding these forms in potentia: to be
able to imagine what is possible, what can become of the present (and past) city.
This chapter is subdivided into three parts: 3.1) “Motion in Space: The Game of
Chess”; 3.2) “Spaces in Motion: Complex Models”; 3.3) “The Center: Lightness and the
5
Calvino has been considered by many critics to be a visual writer (O. Ragusa, M. Belpoliti, among so
many others). As was said before, some critics consider the period in which he wrote his Città to be that in
which his writing was most visual (F. Ricci). These were the years when he was in Paris involved with
Oulipo, a phase consistently been considered a major pivot in his writing career.
11
Spiral”. In exploring and investigating the search for form in space, the concept of
mobility is fundamental. With this in mind, the game of chess will be examined first.
What begins with a rigid grid with limited elements evolves into an exponential process
of unlimited combinations. In the second part, “Spaces in Motion: Complex Models”, the
study will analyze more in depth the mathematical relationships established by a model,
namely the more complex aspects of this essentially binary yet, combinatorial mapping.
This travel through the study of doubles will lead to that of networks and other
intricacies. Finally, in the third part, “The Center: Lightness and the Spiral”, these
complexities will be, if not completely resolved, at least conceptualized as a whole,
through various intrinsically mathematical approaches and perspectives. Departing from
the center of the book, where the theme of lightness shares the central concentric position
with the images of spider-web, suspended, ascending, cities, one can from a distant or
void place (Bauci), also trace spiral shapes, which through opening or closing, lead to the
appearance of evasiveness.
The theme of lightness, which is precisely at the centre of the book, will be a main
point of my research. I intend to analyse how and why Calvino achieves lightness
through the use of mathematical concepts or structures. In this book’s presentation, he
considers that even if most critics have given emphasis to the last line, for him, the
central part of this book is the one that he, as a reader, considers most “luminous”.
Lightness is the main theme, particularly, on this central part where the cities become
“filiforme”.
For Calvino removing the weight of the writing, or lightening the structure, were
crucial aspects; these seem to be interrelated to the repercussions of combinatorial
12
mathematics. Multiplicity leads to reversals (“la molteplicità va attaccata al rovescio”)
thus, giving him the opportunity for constant ironic twists: “L’ironia avverte sempre il
rovescio della medaglia…”, “L’ironia avverte che quello che scrivo va letto con un’aria
un po’ sospesa” (Eremita a Parigi 197).
Last, I’d like to analyse the overall form, which results from the constructions of
this space: its possible meanings and potential interpretations. References to specific
examples within the book, including some interpretations given by critics will be used to
try to find why narrative, as a combinatorial game is essential to Le città invisibili.
Cities and themes emerge, and just as easily seem to disappear: creating the
illusion of a spiral. Within this construction, mathematical concepts stand, at times
obviously expressed as when numbering things, or describing geometric shapes. Yet at
times, mathematical constructs result practically invisible, at time fleeting, No sooner a
space is created than it seems to disappear, just like the cities But Calvino’s mastery
manages to create through mathematics a “filigrana cosi sottile da sfuggire il morso delle
termiti.”
Once again, Calvino’s definition of narrative as a combinatorial game, as well as
Gödel’s Theorem will be applied to analyse why the rules of the game are not enough to
explain all the implications of such a potential system. In the end, the subtlety of what
could have been a rigid grid as well as the flexibility of this “space” is what makes this
book’s fleeting mathematics so alluring.
In conclusion, I intend to demonstrate, through Calvino’s writing, how
mathematics and literature complement each other, how they are interrelated in the search
for new forms, new ideas, new stories. Calvino’s concept of narrative as a combinatorial
13
game (Cibernetica e fantasmi 1968) is the point of departure. The game of ars
combinatoria consists to find or to design new relationships; its challenge, to search for
new forms; its potential, to create spaces that allow freedom of creativity, spontaneity of
imagination, to continue playing.6 Mathematical concepts in Città are a complementary
strategy in the narrative construction of Città, a means or a way for expressing and
acquiring the five indispensably literary values cherished by Calvino: lightness,
quickness, exactitude, visibility, multiplicity, and to each of which he devoted a lecture (a
memo for the reader) in his last legacy, Six Memos (1972).
6
I would like to add a comment appearing on an article by Nobel Prize winner Seamus Heaney on Italo
Calvino. Although written in regards to Palomar, I believe that it could as well have been written on Le
Città invisibili. “But can this tongue that stays so neutrally in its check as it explains in the book’s structural
principles woo us into pleasure and assent all over again in the actual text? Happily, the scheme turns out to
be not just a prescription, what might have been for a lesser imagination a grid acts in this case like a
springboard, and indeed one suspects anyhow that the numerological stuff evolved from the accidents of
composition and not vice versa. Each of these pieces has the feel of a single inspiration being caught as it
rises and then being played for all its life is worth- though not for an instant longer than it takes to exhaust
its first energy “ (“The sensual philosopher Palomar”, Italo Calvino, ed. Harold Bloom, p. 78).
Another critic comments on Calvino’s mathematics (in Palomar): “The passion for geometrical
arrangements, arithmetical operations, symmetries, parallelisms and correspondences plays a pivotal part in
Calvino’s artistic career,” from its early ages. The Cloven Viscount is sustained throughout by a keen even,
slightly perverse, attraction to the concept of geometry” (Dani Cavallaro, 126).
14
Chapter One: Genesis of the Book
“Senza pietre non c’è arco.”
(Le Città invisibili 83)7
I.1. Story of a design
Italo Calvino’s Le città invisibili, based on Marco Polo’s Il Milione, consists in
a collection of travel stories told by Marco Polo to the Kublai Khan, regarding his Empire.
However, all the cities are imaginary. Their dialogues, and Marco’s stories, take place in
the relaxed environment of the magical, exotic garden of the Great Khan, a place full of
temptations that enriches the imagination. In the book’s opening lines, a melancholic
Khan attentively continues to listen with curiosity, even incredulous, to Marco’s fabulous
stories. “Non è detto che Kublai Khan creda a tutto ciò che dice Marco Polo quando gli descrive
le città, ma certo l’imperatore dei tartari continua ad ascoltare il giovane veneziano con più
curiosità e attenzione che ogni altro suo messo o esploratore” (Le città invisibili 5).
In 1960, in a letter to Suso Cecchi D’Amico dated September 2nd Calvino writes
that it is time for him to explain his initial interest in Marco Polo’s stories which later
launched his Città invisibili (“avviato il lavoro del Marco Polo”). In order to “achieve
that minimum excitement of fantasy” (“raggiungere quel minimo d’eccitazione
fantastica”) required for his work, the author confesses, he had to reread Il Milione.
Afterwards, he reveals the reason for this rereading: “to impregnate, soak, absorb himself
with that visionary drive which is the book’s secret” (“per imbevermi di quella carica
visionaria che è il suo segreto.” He then continues:
Insomma, ho cercato di seguire il metodo di Coleridge, che fumando
oppio e leggendo Il Milione compose in sogno “In Xanadu Kublai
7 “Without stones, there is no arch,” is Marco’s reply to the Great Khan as the latter is eager to know what
supports his Empire (82). Marco insists that the individual elements are essential.
15
Khan…”Io non ho oppio a disposizione e non so cosa mi salterà fuori,
comunque mi pare che quello su cui dobbiamo puntare è lo spettacolo
delle meraviglie del mondo come poteva esser concepito in un tempo in
cui il mondo era sconosciuto…(A Suso Cecchi -Roma – from San Remo.
Lettere 657-658).
Regarding the construction of his Città, Calvino points out how it was written
fragmentarily, only one single, isolated piece at a time. In a letter to Claudio Varese,
dated January 20th, 1973, a year after the book had been published, Calvino explains how
he wrote his book “piece by piece”: “il libro è nato pezzo a pezzo per successiva
giustapposizione di pezzi isolati” (Lettere 1193).
The importance of this “fragmentary” method of construction cannot be
understated. These “pieces”, which constitute the individual cities, are the basic units for
the realization of the combinatorial structure the book presents. Written piece by piece,
each city can be used as a “module”, which can later be moved around, interchanged in
‘different arrangements and thus, create different combinations. Not incidentally, within
the book, Marco Polo describes even the ideal city as a model made of fragments: “fatta
di frammenti mescolati col resto, d’istanti separate da intervalli” (Città, 163).8
In a lecture given at Columbia University on March 29th, 1983, Italo Calvino
insists on the fragmentary genesis of this book, how it was written one piece at a time, at
intervals, as if they were poems:“…un pezzetto per volta, a intervalli anche lunghi, come
poesie che mettevo sulla carta, seguendo le più varie ispirazioni. … Così mi sono portato
8
“I will put together, piece by piece, the perfect city, made of fragments mixed with the rest, of instants
separated by intervals, of signals one sends out, not knowing who receives them.” (164)
16
dietro questo libro delle Città negli ultimi anni, scrivendo saltuariamente, un pezzetto per
volta, passando traverso fasi diverse” (Presentazione v).9
These initially “isolated” pieces, described, as “poems” above can be considered
autonomous, complete, and modular. Even as they were created in “successive
juxtaposition”, this modularity allows them to change places, “spostarsi”. This is basic to
combinatorial art.
Moreover, in “Rapidità”, Calvino expresses this predisposition, manifesting a
natural inclination for short stories, a preference shown through his own work: “il mio
temperamento mi porta a realizzarmi meglio in testi brevi: la mia opera è fatta sempre di
“short stories”10 (Lezioni 56).
But, even within Calvino’s short stories, the author points out how the pieces in
Città have a different extent, a distinct compactness that makes them noteworthy: “Ma ho
provato anche componimenti più brevi ancora, con uno sviluppo narrativo più ridotto, tra
l’apologo e il petite-poème-en–prose, nelle Città invisibili….parlo d’una particolare
densità che…ha comunque la sua misura nella singola pagina” (“Rapidità” 56).11
Calvino intended for each page to be a city, a story and vice-versa, before
designing the “map” of this book of imaginary cities. The pieces are consistently
described as isolated brief texts, as poems, of an apparently discontinuous character, like
9
These remarks, originally a lecture given by him at Columbia University on March 1983, have become
the presentation to his book.
10
“By temperament I feel myself more at ease in short pieces: much of my work consists in short stories”
(Six Memos, 49).
11
“But I have experimented with much shorter compositions, with narrative in a smaller scale, with fables
and pètit poèmes-en prose, in my book Invisible Cities… I am speaking of a particular density… that finds
its proper dimension in a single page” (“Quickness” 49).
17
islands, “come uno dei tanti mondi possibili, un’isola in un arcipelago, un corpo celeste
in una galassia” (”Cominciare e finire”154).
The aspect of discontinuity is essential; even Marco’s model of an ideal city is
one based on discontinuity. (Among other examples regarding this recurrent theme, in
“Geografia delle fate”, even the fairies are described as discontinuous: “La loro
apparenza e forse la loro presenza è discontinua” 139).
What Calvino aims at with the petit poème en prose is to achieve that point where
prose and poem meet; what he is searching for is, according to him, again, that form
which is “concise, concentrated” and also “unique, memorable” (“Quickness” 49). “Sono
convinto che scrivere prosa non dovrebbe essere diverso dallo scrivere poesia: in
entrambi casi è ricerca d’una espressione necessaria, unica, densa, concisa, memorabile”
(Saggi I: 671).
Calvino draws another interesting analogy: this time between the mind of the poet
and that of the scientist.“La mente del poeta e in qualche modo decisivo la mente dello
scienziato funzionano secondo un procedimento d’associazione d’immagini che è il
sistema più veloce di collegare e scegliere tra le infinite forme del possibile e
dell’impossibile” (Saggi I: 707).12
His earlier interest in fables, Calvino tells us, was not only due to their style and
rhythm - he finds “economy of expression” to be “the very first characteristic of
fairytale” - but also for logical and structural reasons: “Se in un’epoca della mia attività
letteraria sono stato attratto dai fairytale, dai fairytales non è stato per…ma per interesse
12
“The poet’s mind, and at a few decisive moments the mind of the scientist, works according to a process
of association of images that is the quickest way to link and to choose between the infinite forms of the
possible and the impossible” (Six Memos 91).
18
stlistico e strutturale, per l’economia, il ritmo, la logica essenziale con cui sono
raccontate” (Lezioni 43-4).13 Ironically, he quotes Charles Perrault: “Les fèes n’étaient
longues a leur besogne.”
Again, the emphasis is on being concise, brief, and swift as in an elegant
mathematical demonstration:“Il suo segreto sta nella economia del racconto; gli
avvenimenti, indipendentemente dalla loro durata ,diventano puntiformi, collegati da
segmenti rettilinei, in un disegno a zigzag che corrisponde a un movimento senza sosta”
(“Rapidità” 43).14
Regarding the concept of “punctiform”, or, having the form of a point, informally
means having no dimension at all. However, a point is the basic unit in geometry. It can
indicate a location, it can be moved around, by simply changing its coordinates. But also,
by combination with one or more points, it is capable of forming lines, of all kinds, as
well as figures or shapes of multiple dimensions. This is analogous to the concept that
Calvino says he aspires in his short stories. “Keeping it short”, (in line with Oulipo),
became a strategy for Calvino. Each piece contains the ability to move, to explode as a
point of departure, to combine.
Besides his interest in short forms and fables, this fragmentary construction is not
casual. “Il mio temperamento mi porta allo “scrivere breve” e queste strutture mi
permettono d’unire la concentrazione nell’invenzione e nell’espressione con il senso delle
13
“If during a certain period of my career as a writer I was attracted by folktales and fairytales…it was
rather because of my interest in style and structure, in the economy, rhythm and hard logic with which they
are told” (Six Memos 35).
14
“The secret of story telling lies in its economy: the events, however long they last become punctiform,
connected by rectilinear segments, in a zigzag pattern that suggests incessantmotion” (35).
19
potenzialità infinite” (Lezioni 131).15 In this manner he acquires the necessary elements
for the mathematical combinations that will be the main tool to use on his bookcomposed of imaginary cities. “Ma direi che oggi la regola dello ‘scrivere breve’ viene
confermata anche dai romanzi lunghi, che presentano una struttura accumulativa,
modulare, combinatoria” (Lezioni 131).16
This process seems to confirm, so far, that the possible applications of
combinatorial mathematics in literature are based predominantly on discrete mathematics.
(However, as every concept in this book, its opposite, that is continuity, will be part of
the mathematical model. This perspective will be amplified as we examine the topics of
exactitude and multiplicity, where the problem of organizing and interconnecting the
pieces will be explored.) The relevance of establishing the basics of a combinatorial
system lies, as we will examine, in the potentiality of the process.
Each fragment, as said before is a description. Describing anything, Calvino says,
seems quite simple at first, but later turns out to be much more problematic. Here, once
more, the author takes a rather mathematical approach: regarding the intricacies of
writing a description, Calvino emphasizes on the importance of grasping the essential:
“L’importante naturalmente è cogliere l’essenziale” (Album Calvino 255).
Again, each of these descriptions constitutes a single city. In his essay “Gli dei
della città”, as Calvino explains the process of seeing or rather observing a city (which, in
this case is imaginary) he insists that to look into a city requires much more than opening
15
“My temperament prompts me to “keep it short, and such structures as these enable me to unite density
of invention and expression with a sense of infinite possibilities” (Six Memos 120).
16
“But I would say that today the rule of “Keep it short” s confirmed even by long novels, the structure of
which is accumulative, modular and combinatory” (Six Memos 120).
20
one’s eyes. He describes a process, which could as well be that of solving an algebraic
problem; first, eliminate, then simplify, in order to reduce to the essential and then join
the pieces into an analytic design, a whole unit.“Per vedere una città non basta tenere gli
occhi aperti. Occorre per prima cosa scartare tutto ciò che impedisce di vederla…poi
occorre saper semplificare, ridurre all’essenziale l’enorme numero d’elementi che a ogni
secondo la città mette sotto gli occhi di chi la guarda, e collegare i frammenti sparsi in un
disegno analitico e insieme unitario…” (“Gli dei della città”, Saggi I: 340).
The concepts of space and form, and their relationship to time and movement
become the essential aspects for a description. In fact, in his essay “Savona: storia e
natura”, Calvino explicitly describes this process:
Se si vuole descrivere un luogo, descriverlo completamente , non come
un’apparenza momentanea ma come una porzione di spazio che ha una
forma, un senso e un perché, bisogna rappresentarlo dalla dimensione del
tempo, bisogna rappresentare tutto ciò che in questo spazio si muove ,
d’un moto rapidissimo o con inesorabile lentezza: tutti gli elementi che
questo spazio contiene o ha contenuto nelle sue relazioni, passati, presenti,
e future . Cioè una vera descrizione d’un paesaggio finisce per contenere
la storia di quel paesaggio, dell’insieme dei fatti che hanno lentamente
contribuito a determinare la forma con cui esso si presenta ai nostri occhi,
l’equilibrio che si manifesta in ogni suo momento tra le forze che lo
tengono insieme e le forze che tendono a disgregarlo (“Savona: storia e
natura”. Saggi II: 2390).
In essence, the author defines a place as “portion of space which has a form”.
Motion and time are also important. In fact, motion is really a change in space during
time. In the end, these essential elements, interrelated, must be present in a description,
which becomes the story of the place. A story that, in turn, contains all the elements that
gave this space its form: a form which is established by an equilibrium founded by these
basic elements – space, time and motion- and any constructive or destructive force, which
tends to erase, or change the form itself. (In Città, the Khan reflects on analogous aspects:
21
“Al contemplarne questi paesaggi essenziali, Kublai rifletteva sull’ordine invisibile che
regge le città, sulle regole cui risponde il loro sorgere e prendere forma e prosperare e
adattarsi alle stagioni, e intristire e cadere in rovina” (122).
The emphasis on space and its relationship to time in regards to a description,
appears again as a hypothesis in an essay entitled “Ipotesi di descrizione di un
paesaggio”: “Dunque è naturale che una descrizione sia scritta una operazione che
distende lo spazio nel tempo…perciò una descrizione di paesaggio, essendo carica di
temporalità è sempre un racconto“ (Saggi II: 2694). The initial hypothesis leads to the
conclusion that a description is itself a story.17
I. 2. From Description to Structure
In his Lezioni americane, Calvino reveals how he found himself oscillating
between writing descriptions and designing structures: “Così negli ultimi anni ho
alternato i miei esercizi sulla struttura del racconto con esercizi di descrizione…”(83).18
His writing, alternated back and forth: from structure to description or from description to
narrative structure, once he felt that he had completely investigated all the alternatives of
one or the other”: “…oscillo continuamente e quando sento d’aver esplorato al massimo
le possibilità dell’una mi butto sull’altra e viceversa” (83).19
In the case of Città, only after the city-descriptions were completed, did the
writing proceed with their organization. Thus, the author was confronted with the
17
More examples appear on “Altre Descrizioni” (Saggi II: 2681-2699).
18
“I have alternated my exercises in the structure of the story with other exercises in description…”
(“Exactitude” 74)
19
“… continuously switching back and forth […] and when I feel I have fully explored the possibilities of
one, I rush across to the other and vice versa” (“Exactitude” 75).
22
problem of giving an order to the individual components of the book. After paying close
attention to each individual city-description, the writer had to distance himself. In order
to be able to see the whole, he needed to find another perspective, perhaps a more
abstract point of view, so as to be able to differentiate and draw relationships among the
components: “la forma delle cose si distingue meglio in lontananza” (99).20
As the author unveils, each page or city had been assigned to a series; a series that
at times had to be distributed among other categories. Thus he had to solve the problem
of organization, by classifying, looking for new definitions, and regrouping, until he was
able to visualize the form and the meaning he wanted for his book.
Creating an arrangement in which to place each one of the cities implies drawing
a map. This map defines the form of the narrative space. A map is a visual guide.
Mathematically, mapping is considered a function between spaces. Also, a map can be
viewed as a correspondence between the elements of two sets. Here, the two spaces or
sets would be the cities and the book. In the case of a book, the map becomes the
itinerary for the narrative. This is the purpose of the index.
When the mapping is “one to one”, each element on the first set is assigned one
element from the second. But mapping can be more complex, where one element can
correspond to many others or vice-versa. Such complexity is what Calvino attempts to
resolve and demonstrate in Città, through the potential strategy of combinatorial
mathematics. 21
20
21
“… the form of things can be disverned better at a distance” (98).
Many references to maps, to the Khan’s atlas can be found throughout the book, as will be explored on
the next chapter.
23
Multiple maps appear throughout the book. When describing the city of Fedora,
for instance, Marco Polo, addressing himself directly to the Great Khan, explains that the
map of the Emperor should allocate all possible cities: “Nella mappa del tuo impero, o
grande Khan, devono trovare posto sia la grande Fedora di pietra sia le piccole Fedore
nelle sfere di vetro…Non perché tutte [sono] ugualmente reali; ma perché tutte [sono]
solo presunte”(31-32).22
This map of cities which are “only assumptions”, since they are imaginary, is
what we find in Calvino’s book. (Cartography is also a recurrent theme in Calvino’s
writing, including his essays, for example “Il viandante nella mappa” 426-433). Also, in
the last chapter, the topic of the Khan’s atlas is constantly repeated.23 ”Il Gran Kan
possiede un atlante dove tutte le città dell’impero e del reami circonvicini sono disegnate
palazzo per palazzo e strada per strada, con le mura, i fiumi, i punti, i porti, le scogliere”
(137).24
Il Gran Kan possiede un atlas i cui disegni figurano l’orbeo terracqueo
tutt’insieme e continente per continente, i confini dei regni più lontani, le
rote delle navi, i contorni delle coste, le mappe delle metropoli più illustri
e dei porti più opulenti (138).25
Il Gran Kan possiede un atlante in cui sono raccolte tutte le mappe di tutte
le città, quelle che elevano le loro mure su salde fondamenta, quelle che
22
“On the map of your empire, O Great Khan, there must be room for both the big, stone Fedora and the
little Fedoras in glass globes. Not because they are all equally real, but because they are only assumptions”
(32).
23
This will be examined within the frame.
24
“The Great Khan’s owns an atlas where all the cities of the empire and the neighboring realms are drawn,
building by building, and street by street, with walls, rivers, bridges, harbors, cliffs” (135).
25 “The
Great Khan owns an atlas whose drawings depict the terrestrial globe all at once and continent by
continent….” (136)
24
caddero in rovina e furono inghiottite dalla sabbia, quelle che esisteranno
un giorno e a cui posto ancora non s’aprono che le tane delle lepre (139).26
Again and again this theme reappears, even in the closing part of frame, at the book’s
end: “L’ atlante he questa qualità: rivela la forma delle città che ancora non hanno una
forma né un nome” (140).27 With respect to form, the author presents his book in terms of
a geometrical figure, “fatto a poliedro” which will be further discussed in Chapter 3.28
In his article “Cominciare e finire”, Calvino exposes the task of having to choose,
among the many alternatives, one with the potential that not only serves its structural
purposes, but also maintains the pleasure, the joy of creativity. In Città, Calvino, chose
the image of the city and was able to build a “multifaceted structure” and a “network in
which one can follow multiple routes and draw multiple ramified conclusions” (Six
Memos 71). His intention was to find the “best” structure, one, which would allow the
various series to alternate, to interweave among each other. “È sulla base del materiale
che avevo accumulato che ho studiato la struttura migliore, perché volevo che queste
serie si alternassero, si intrecciassero…” (Città vii).
In “Exactitude”, the third of his five lectures in Lezioni americane, (Six Memos),
Calvino not only states that this was the book” in which he managed to say the most”, but
in order to explain why, he provides two double reasons.
26
“The Great Khan owns an atlas in which are gathered the maps of all the cities: those whose walls rest on
solid foundations, those who fell in ruins and were swallowed up by the sand, those that will exist one day
and in whose place now only hares’ holes gape.” (137-8)
27
28
“The atlas has this quality: it reveals the forms of cities that do not yet have a form or name.” (138)
A polyhedron consists of a “solid figure, or its surface, that is bounded by four or more polygonal faces
in such a way that pairs of faces meet along the edges, and three or more edges meet in each vertex. In turn,
a polygonal face, or polygon, is a closed plane figure bounded by three or more straight-line segments that
terminate in pairs at the same number of vertices and do not intersect other than at their vertices” (E.J
Borrowsky and J.M. Borwein).
25
First, the image-symbol of the city constitutes a potential source of possibilities
for his Città. Each city is a point of departure, a reference, not only of that particular city
but also of the city in general. According to him, with a single image, that of the city, he
was able to express “the tension between geometric rationality and the entanglements of
human life” (chaos). “Un simbolo più complesso, che mi ha dato le maggiori possibilità
di esprimere la tensione tra razionalità geometrica e groviglio della esistenza umana è
quello della città” (Lezione ottanta).29 (An analogous tension is found in the book
between the rationality of the Kublai Khan and Marco Polo’s insistent storytelling of the
cities.)
Secondly, the author found that he could not only build a ”multifaceted structure”
but also “a network in which one can follow multiple routes and draw multiple, ramified
conclusions” (Six Memos 71).
Il mio libro in cui credo d’aver detto più cose resta Le città invisibili
perché ho potuto concentrare su un unico simbolo tutte le mie riflessioni,
le mie esperienze, le mie congetture, e perché ho costruito una struttura
sfaccettata in cui ogni breve testo sta vicino agli altri in una successione
che non implica una consequenzialità o una gerarchia ma una rete entro la
quale si possono tracciare molteplici percorsi e ricavare conclusioni
plurime e ramificate (Lezioni 80).30
For Calvino, exactitude is mostly defined by three elements. The first one, “a well
defined and well calculated plan for the work in question” is what we are examining at
this point. The second, “an evocation of clear, incisive, memorable images, in Italian…
icastico, from the Greek”, is found in the image of the city or in each individual city, and
29
“A more complex symbol, which has given me greater possibilities of expressing the tension between
geometric rationality and the entanglements of human lives, is that of the city” (“Exactitude” 71).
30
“The book in which I managed to say the most remains Invisible Cities, because I was able to concentrate
all my reflections, experiments and conjectures on a single symbol; and also because I built up a multifaceted structure in which each brief text is close to the others in a series that does not imply logical
sequence or hierarchy, but a network in which one can follow multiple routes and draw multiple
conclusions” (“Exactitude” 71).
26
we will also examine the visual images; the third, “a language as precise as possible both
in choice of words and in expression of the subtleties of thought and imagination”, will
be examined in the language, words and expressions provided by mathematics in the
cities as we investigate. “Esattezza vuol dire per me soprattutto tre cose: 1) un disegno
dell’opera ben definito e ben calcolato; 2) l’evocazione d’immagini visuali nitide,
incisive, memorabili…”icastico”; 3) un linguaggio il più preciso possibile come lessico e
come resa delle sfumature del pensiero e dell’immaginazione” (Lezioni 65).31
1.3. Design of a Story: The Index
Ti ripeto la ragione per cui te la descrivevo: dal numero delle città
immaginabili occorre escludere quelle i cui elementi si sommano senza un
filo che li connetta, senza una regola interna, una prospettiva, un discorso.
È delle città come dei sogni: tutto l’immaginabile può essere sognato, ma
anche il sogno più inatteso è un rebus che nasconde un desiderio, oppure il
suo rovescio, una paura, anche se il filo del loro discorso è segreto, le loro
regole assurde, le prospettive ingannevoli, e ogni cosa ne nasconde
un’altra (Marco Polo to Kublai Khan. Città 43-44).32
Regarding the organization of the accumulated fragments, Calvino also comments
how all these pieces did not yet constitute a book. For him, a book is a space, but a space
that must have a plot, an order and a way out. In order for the written pages to become a
book, the book must have a construction, an itinerary.
31
“To my mind exactitude means three things after all: 1) a well defined and well calculated plan for the
work in question; 2) an evocation of clear, incisive, memorable images; in Italian we have an adjective that
doesn’t exist in English, “icastico”, from the Greek; 3) a language as precise as possible both in choice of
words and in expression of the subtleties of thought and imagination” (“Exactitude” 56).
32
“I shall explain the reason why I was describing this to you: from the number of imaginable cities we
must exclude those whose elements are assembled without a convincing thread, an inner rule, a perspective,
a discourse. With cities, it is as with dreams: everything imaginable can be dreamed, but even the most
unexpected dream is a rebus that conceals a desire or, its reverse, a fear. Cities, like dreams are made of
desires and fears, even if the thread of their discourse is secret, their rules absurd, their perspective
deceitful, and everything conceals something else” (Cities 44).
27
Accordino to Calvino, a book is "qualcosa con un principio e una fine". That is, it
must have a beginning and an end. A book is thus a “space”, one that allows the reader to
enter and wander through it: "uno spazio in cui il lettore deve entrare, girare, magari
perdersi." However, the author insists, a book must also have “una costruzione, cioè si deve
poter scoprire un intreccio, un itinerario, e una soluzione [...]" (Città vi).33 An analogous
idea, referring to what a story should entail, appears in his article “L’orecchio, il cacciatore,
il pettegolo”: “Il raccontare è l’operazione per cui tra i dati innumerevoli che formano il
tessuto continuo delle vite umane, se ne sceglie una serie in cui si suppone un senso e un
disegno: indici e tracce appunto di una storia con un principio e una fine, d’un percorso
esistenziale determinato, d’un destino” (103).
Accordingly, a story presupposes selectiveness and a structure. “Il
racconto…propone insieme singolarità e geometria: si dà racconto quando la singolarità
dei dati si compone in uno schema, sia esso rigido o fluido”.34 (“L’orecchio, il cacciatore,
il pettegolo” 104). The emphasis is on singularity: “Riconoscimento della singolarità che
sfugge al modello normativo; costruzione di un modello più sofisticato, tale da aderire a
una realtà più accidentata e spigolosa; nuova rottura delle maglie del sistema; e così via”
(Enciclopedia 103).
His Città presents a unique construction of this space - a fabulous Empire of
imaginary cities - designed by his use of ars combinatoria. This “itinerary” is what the
index displays. In fact, the author considered this book to be “molto oulipiano, soprattutto
per il suo indice”. This gives us an important point of departure. The oulipian character
33
34
This Presentation was originally a lecture given at Columbia University on March 29th, 1983.
Here, the word “schema” (from Greek) can mean shape, plan or model. Calvino points out how a scheme
can be rigid or the opposite, “fluid”, that is, it can change form.
28
refers to the multiplicity of potential arrangements suggested, to the potential unfolding
of possible alternatives that emerge from the undeniably combinatorial design of the
index.
The reader’s initial encounter is with the external organization of the book, as
presented by the index. However, this particular index of Città, originally placed by the
author at the beginning of the book, instead of at the end, as in his other books, presents a
rather particular – mathematical - arrangement. .
Within an index the reader expects to find a map, an itinerary to “cruise” through
the book as well as the fabulous Empire of Kublai Khan; that is, to “read” the cities and
“travel” through the book. An itinerary, unlike a static map, implies travel, a change in
location, that is, motion. Yet, at first glance, the external organization displays an
apparently random arrangement insinuating a complete lack of method. Insidiously, this
apparent absence of order only manages to make the reader suspicious, or curious as if
confronting a hidden code. All of a sudden, what was expected to be a means for
orientation becomes disorienting.
A closer look gives an alert, a clue as to the maneuvers of the author. Upon
observation, some subtle yet intricate patterns that suggest other possibilities reveal
themselves within the design. The reader, like the Khan, needs to try “to discern…the
tracery of a pattern so subtle it could escape the termites’ gnawing” (“…discernere…la
filigrana d’un disegno così sottile da sfuggire al morso delle termite” (1).
Calvino’s combinatorial art does not limit itself to the variations of the city image
or to the ‘mapping’ of the city stories but extends itself to become a blueprint for the
29
architecture of the book. Not only do these mathematical combinations guide the spatial
arrangement but they also constitute the game plan.
The book contains fifty-five cities divided in nine chapters. The first and last
contain ten cities whereas the remaining seven chapters contain five cities each. Eleven
themes are used, each paired with five cities. (This pairing further emphasizes the binary
character of the book.) Each city has its own story, the title of which is assigned
according to the theme associated and a number. The combinations by which each city
story is designated, not just a place, but also a title, become “X and the City”, where X
equals one of the eleven themes: memoria, desiderio, segni, sottili, scambi, occhi, nome,
morte, cielo, continue, nascoste).
In the scheme indicated by Calvino, each of the nine sections is subdivided in
parts according to a combinatorial system. The rules of the narrative game are the basis of
the numbers assigned to each city or group of cities associated by the combinatorial
arrangements. Each theme appears five times, with five different cities, ordered as
expected by the pattern.
The text is ordered in a way mathematically equivalent to an algorithm. The idea
behind is simply to create a matrix, which in turn, with a limited number of elements, is
able to generate (and simultaneously include) a multiplicity of variations. An algorithm is
nothing but a step-by-step procedure by which a problem can be solved by a set of rules
and a clear stopping point. Also, it can be defined as a recursive method, in which each
element of a sequence is derived from its preceding ones. Problems dealing with the
complexities of combinatorial structures lead to the use of algorithms. Within Oulipo, an
algorithm became a method, which was well incorporated into their texts. “Occorre dire
30
che nel metodo dell’Oulipo…l’opera non è un esempio delle potenzialità raggiungibili
solo attraverso la porta stretta di quelle regole…Ogni esempio di testo costruito secondo
regole precise apre la molteplicità “potenziale” di tutti i testi virtualmente scrivibili
secondo quelle regole, e di tutte le letture virtuali di questi testi” (“La filosofa di
Raymond Queneau”, Saggi II: 1429).
Potentially, an algorithm provides a creative tool to compose, invent a
proliferation of possibilities, but also, for a reversal process, that is, to decompose, to
analyze, to reduce. The ordering of the book unveils a combinatorial distribution whereby
themes and numbers (each one assigned to a specific city) are arranged in such a fashion
as to allow them to appear, alternate and just as easily disappear.
This arrangement, formed by the relationships between themes and numbers,
forms a thematically and structural “network” (as described in “Exactitude”). The
sequence of cities forms a pattern that, at first, builds itself by counting up, then dissolves
by countdown. Eleven themes are each combined with five cities as follows: 1 12 132
14321 54321 54321 54321 54321 54321 54321 54321 5432 543 54. 5). This sequence
can be better visualized, when graphically drawn, as a parallelogram, a figure that in
three dimensions resembles a spiral.
1
21
321
4321
54321
54321
54321
54321
54321
54321
54321
5432
31
543
54
5
The numbers designates the five appearances of each of the eleven themes. After their
fifth appearance, they each one disappear, as expected by the recursive pattern of the
algorithm. The mastery of this strategy lies in the subtle way in which themes and
structure evolve together in the narrative.
This construction and its self-erasure suggest a ‘spiral’ design and successfully
convey a sense of dynamic instability. In certain systems, movement in opposite
directions sometimes provoke self-erasure: a phenomenon which begins by constructing
itself one way, then reverses its movement, undoing itself by undoing its path. Through a
quasi- magical procedure, things appear and then vanish leaving no apparent trace.
Similarly, this image not only occurs frequently throughout the cities, but also is overtly
exemplified by the spiral shape of the book, where all cities eventually seem to disappear.
“Le immagini della memoria, una volta fissate con le parole si cancellano, - disse Polo.
Forse, Venezia ho paura di perderla tutta in una volta, se ne parlo. O forse, parlando
d’altre città, l’ho già perduta a poco a poco.” (88)
The sense of mobility, in turn, is due to the modular character of its structure; that
is, a system formed by units 35that can be moved around and arranged in a variety of
ways, to “build a multifaceted structure.”
The desired effect of malleability achieved by the author draws as much attention
to their present arrangement as to other options. The index does not have the appearance
35“Memory’s images, once they are fixed in words, are erased," Polo said. "Perhaps I am afraid of losing
Venice all at once,if I speak of it. Or perhaps, speaking of other cities, I have already lost it, little by little.”
(87)
32
of a static table of contents; its unusual sequence of numbers, points towards one of the
many possible ‘routes’ available. The intention is to show other possibilities besides the
one presented, so that one’s interest becomes to be actively involved in the game of
finding new combinations, new connections through this “network”, in which, as the
author intended, “one can follow multiple routes and draw multiple ramified conclusions”
(Six Memos 71).
By unmasking the combinatorial process, Calvino creates the illusion of a book
under construction (an illusion emphasized by the frame.) One becomes increasingly
aware that this book, as a modern city, can be entered from any point. In this city-book,
many entrances as well as exits exist. The map provided by the index can potentially be
used to generate multiple maps; each one starts a new game.“La struttura è libertà,
produce il testo e nello stesso tempo la possibilitá di tutti i testi virtuali che possono
sostituirlo. Questa è la novità che sta nell’idea della molteplicità potenziale implicita
nella proposta di una lettura che nasca delle costrizioni che essa sceglie e s’impone“ (“La
filosofia di Raymond Queneau” , Saggi I: 1429).
The reading may favor one or another: to read them in the sequence in which they
appear or by categories, or by their respective numberings; even by a combination of any
of these that may prove to have further elements in common or in contrast. The reading
thus proceeds with multiple intersections, between the numbers of the category and the
number of its location within the text. Potentially, this process generates infinite
combinations; combinatorial mathematics provides the instrument to think of all possible
cases in advance. When applied to literature, narrative becomes a combinatorial game. In
33
“I mille giardini”, an analogous strategy is used to tell the story of a fantastic garden in
Kyoto:
Ogni pietra corrisponde a un passo, e a ogni passo corresponde un
paesaggio studiato in tutti I dettagli, come un quadro: il giardino è stato
predisposto in modo che di passo in passo lo sguardo incontri prospettive
diverse, un’armonia diversa nelle distanze che separano…lungo il
percorso lo scenario cambia completamente molte volte…il giardino si
moltiplica in innumerevole giardini (Collezione di sabbia: 187).
While trying to define exactitude, Calvino expresses how this search for
measured, limited forms, inevitably leads to the concept of infinity. Perhaps here we can
trace another reason why mathematics becomes useful to him. To be able to express the
sense of infinite potential is simplified in mathematics by the use of recursive methods,
such as the algorithm presented by his index.“Volevo parlarvi della mia predilezione per
le forme geometriche, per le simmetrie, per le serie, per la combinatoria, per le
proporzioni numeriche, spiegare le cose che ho scritto in chiave della mia fedeltà all’idea
di limite, di misura …Ma forse e proprio questa idea che richiama quella di ciò che non
ha fine: la successione dei numeri interi, le rette di Euclide…” (Lezioni 76-7).36
I. 4. Design of a Story: the Frame
The frame is another key element in the construction of the book, other than
“enclosing” the book; it acts as a link or thread to the collection of stories.
The nine chapters in which the book is divided begin and end with a dialogue
between Marco Polo and Kublai Khan. At the opening and closure of each of the nine
sections, these “dialogues” provide commentaries and reflections regarding the stories
36 “I wanted to tell you of my fondness for geometrical forms, for symmetries, for numerical series, for all
that is combinatory, for numerical proportions; I had written in terms of my fidelity to the idea of limits, of
measure…But perhaps it is this idea of forms that evokes the idea of the endless: the sequence of whole
numbers, Euclid’s straight line” (Six Memos 68).
34
being narrated. They give the illusion that the book is being discussed, even constructed
while it is being read. The frame serves the purpose of suspending the story in two ways:
as a thread and as a periodical pause.
Due to the proliferation of possibilities inherent to the combinatorial game, the
writer (and eventually the reader) must find ways to deal with this multiplicity. The
potentiality of the system becomes a challenge. The author, who must handle this
multiplicity, uses various techniques for the construction of the book. Besides the indexa combinatorial algorithm- the frame by which Calvino links and encloses his stories,
involves other key strategies for the design of the book.
As mentioned, each of the nine parts of the book opens and closes with a text in
italics consisting of an ongoing yet constantly suspended sequence of dialogues
According to the author, these dialogues truly are his own reflections, conceived as he
was writing the individual stories “come commenti di Marco Polo e del Kan e queste
riflessioni tiravano ognuna dalla sua parte”. As he continued letting them develop on their
own, Calvino devised a construct that would evolve parallel to the narrative:” Così ho
avuto un insieme di materiale che ho cercato di far correre parallelamente al resto”. Thus,
he contrived this montaggio where the main interaction between the interlocutors
suspends and then recaptures itself at the beginning and end of each chapter.37
Coincidentally, the form designed by the index organization can be visualized as a
parallelogram, as we just examined. A parallelogram is a figure in which the opposite
sides are parallel; that is parallel lines frame this shape, the parallelogram, just as in the
37
According to Calvino, the frame of Città is truly a commentary to the book, showing how it is discussed
and questioned, as it is written- constructed (Città viii-ix).
35
book. Similarly, the parallel lines of these dialogues frame the stories. However, the
interpretations of the overall structure will be the last part of this research.
Hence, the book appears to display its own self-commentary in the sense that
Marco, as the storyteller (role of the writer), and the Emperor, as the listener (role of the
reader), continue to have their own thoughts and discussions in between the storytelling.
The frame serves as a reflection to the stories. “Nella mente del Kan l’impero si rifletteva
in un deserto di dati labili e intercambiabili come grani di sabbia da cui emergevano per
ogni città o provincia le figure evocate dai logogrifi del veneziano” (22).38 As the book
reflects upon itself within the frame, the latter can be viewed as a mirror. In fact, the
image of the mirror is used by Marco to further explain his travel stories: “L’altrove è
uno specchio in negativo. Il viaggiatore riconosce il poco che è suo, scoprendo il molto
che non ha avuto e che non avrà” (27).39
Mirror images, as will be examined in the next chapter, become recurrent themes
within the cities. (A mirror image is a one-to-one mapping, but the image is reversed, and
virtual.) The best example would be Valdrada, the mirror city par excellence: “Non
esiste o avviene cosa nell’una Valdrada che l’altra non ripeta, perché la città fu costruita
in modo che ogni suo punto fosse riflesso dal suo specchio” (53).40
From the opening lines of the book, the Khan’s disbelief in Marco’s stories is not
enough to keep him from listening with attentive curiosity, trying to find within the
38 “In
the Khan's mind the empire was reflected in a desert of labile and interchangeable data, like grains
of sand, from which there appeared, for each city and province, the figures evoked by the Venetinan's
logogriphs” (22).
39
And Marco’s answer was: “Elsewhere is a negative mirror. The traveler recognizes the little that is his,
discovering the much he has not had and will never had”. (29)
40 “Nothing
exists or happens in the one Valdrada that the other Valdrada does not repeat, because the city
as so constructed that its every point would be reflected in its mirror” (53).
36
stories “the tracery of a pattern so subtle it could escape the termites’ gnawing”: “Solo
nei resoconti di Marco Polo, Kublai Kan riusciva a discernere, attraverso le muraglie e le
torri destinate a crollare, la filigrana d’un disegno così sottile da sfuggire il morso delle
termiti.”41 Marco, understanding that the Emperor was trying to follow his train of
thoughts, begins to unveil the purpose of his travels and advises him “to sharpen his
eyes”. “Il fine delle mie esplorazioni è questo: scrutando le traccie di felicità che ancora
s’intravedono,ne misuro la penuria. Se vuoi sapere quanto buio hai in torno, devi
aguzzare lo sguardo sulle fioche luci lontane” (59).42
The intention of these discussions and reflections is, once again, as in the index,
to reveal a book in the making. In Calvino’s own words, “il libro si discute e si interroga
mentre si fa…”(Città ix). These commentaries seem to dismantle, to unmask the world of
the writer.
For instance, at the end of chapter five, the central one of the book, we find the
following dialogue, the shortest piece of his frame pieces:
Marco descrive un ponte pietra per pietra.
- Ma qual è la pietra che sostiene il ponte? – chiede Kublai Kan.
- Il ponte non è sostenuto da questa o quella pietra – risponde Marco, ma dalla linea dell’arco che esse formano.
41
The Emperor’s particular interest in Marco’s stories seems justified by his eagerness in trying to find a
‘subtle design’’, (a filigrana), a narrative thread in the labyrinth of cities. “Non è detto che Kublai Kan
creda a tutto quell che dice Marco Polo quando gli descrive le città visitate nelle sue ambascerie, ma certo
l’imperatore dei tartari continua ad ascoltare il giovane veneziano con più curiosità e attenzione che ogni
altro so messo o esploratore” (7).
42
This happens, on the fourth chapter, when the Emperor’s disbelief turns into frustration, as he confronts
Marco questioning him regarding the existence of his cities: “Le città tue non esistono. Forse non sono mai
esistite. Per certo non esisteranno più [.…] Perché menti all’imperatore dei tartari, straniero?”(59).
37
Kublai Kan rimane silenzioso, riflettendo. Poi soggiunge: Perché mi parli
delle pietre? E solo dell’arco che m’importa.
Polo risponde: - Senza pietre no c’è arco (83).43
The process of construction appears to be, at least partly, unveiled as if the author
wanted to make his reader his accomplice in the creation of cities. Such apparent
confidentiality, almost whispered to the reader, becomes a temptation to continue
reading, eager for another hint of towards a full disclosure: “Dimmi ancora un’alta città”,
insists the Emperor, at the opening of chapter six, as he understood Marco’s revelation on
the importance of the stones, in analogy to the cities.
The “construction” process, of which we suspiciously become participants,
becomes a journey into the unknown, where nothing is absolutely certain and everything
implies a search. As the first chapter ends, Kublai continues to look for a pattern, but the
connecting thread remains uncertain”: “Il Gran Kan decifrava I segni, però il nesso tra
questi e i luoghi visitati rimaneva incerto….” (22). Again, at the end of the second
chapter, the relationships were still confusing: “Non sempre le connessioni tra un
elemento e l’altro del racconto risultavano evidenti all’imperatore…” (39).
43 Marco Polo describes a bridge, stone by stone.
"But which is the stone that supports the bridge?"Kublai Khan asks.
"The bridge is not supported by one stone or another,"Marco answers, "but by the line of the arch that they
form "
Kublai Khan remains silent, reflecting. Then he adds: "Why do you speak to me of the stones? It is only the
arch that matters to me."
Polo answers: "Without stones there is no arch." (82).
38
The enchantment of Città invisibili consists in granting the reader the chance to
become accomplice of this creative process, giving him the freedom to visualize the
invisible. Departing from brief descriptions, and their commentaries, the reader feels free
not just to travel through the book, but also to visually design or redesign each story, to
inhabit each one entirely in his imagination.
Analogously, at the opening of the third chapter, before the dialogue resumes, the
Great Khan infers that, going from city to city, involved, not travelling, but a “change of
elements”. From that point, his ““mind sets out on its own, and after dismantling the city
piece by piece, he reconstructed it in other ways, substituting components, shifting them,
inverting them” (43).44
Simultaneously, Calvino wraps not just the cities but also the reader in the
dialogue between Kublai Khan and Marco Polo; a dialogue, which gives the collection of
cities coherence, a meaning, perhaps, or an “explanation” for their existence.
The stories enclosed, these petits poèmes en prose, these architectural dreams, are
linked by this frame, which serves as a bridge between them. Their magic secret partly
lies in their economy (brevity and conciseness), as previously discussed. Within the
frame, their power is described as emblematic, as they seem to leave indelible marks in
Kublai’s (and the reader’s) mind: “Ma palese o oscuro che fosse, tutto quell che Marco
Polo mostrava aveva il potere degli emblemi, che una volta visti non si possono
dimenticare né confondere” (22). (However, it should be noticed, this fact does not
prevent him from becoming aware that the cities resemble each other.)
44
“Adesso da ogni città che Marco gli descriveva, la mente del Gran Kan partiva per suo conto, e smontata
la città pezzo per pezzo, la ricostruiva in un altro modo, sotituendo ingredienti, spostandoli, invertendoli”
(43).
39
Even if the way in which the stories follow one another can be altered, the speed
with which these ‘little mirrors’ seem to appear and disappear, reflect, refract and reverse
themselves is inevitable. With “agility, mobility and ease”, they spring “from one subject
to another”, “lose the thread a hundred times” and “find it again after a hundred more
twists and turns”, “all qualities that go with writing where it is natural to digress”. 45 This
flowing rhythm, this speed of the fleeting city stories is both set in motion and dilated by
the frame.
Within this apparently timeless space of the cities, this cornice keeps the tempo as
if it were a pendulum. As such, due to its dual alternation, other than opening and closing
each chapter, this ongoing dialogue is strategically linked to the central theme of
duplicity (which leads to further multiplications).
In fact, the author reveals how everything in this particular book becomes
doubled: “Nelle Città invisibili ogni concetto e ogni valore si rivela duplice anche
l’esattezza” (Lezioni 80).46 Furthermore, he confesses, this duality is present throughout
his writings: “In realtà sempre la mia scrittura si é trovata di fronte due strade divergenti
che corrispondono a due diversi tipi di conoscenza: una che si muove nello spazio
mentale d’una razionalità scorporata, dove si possono tracciare 'linee che congiungono
punti, proiezioni, forme astratte, vettori di forze; l’altra che si muove in un gremito
d’oggetti …” (82).47
45
This definition, given by Calvino on “Quickness” echoes his own definition of a book as a space (Six
Memos 48).
46
”In my Invisible Cities every concept and value turns out to be double- even exactitude” (“Exactitude”
72).
47
“The fact is, my writing always found itself facing two divergent paths that correspond to two different
kinds of knowledge. One path goes into the mental space of bodiless rationality, where one may trace lines
40
These divergent paths are represented in the book by the Khan’s rationality and
Marco’s inexhaustible story telling. The eighth chapter closes and ends by repeating and
emphasizing this important divergence of thought between the characters:
Il Gran Kan cercava d’immedesimarsi nel gioco: ma adesso era il perché
del gioco a sfuggirgli, ma di cosa? Qual era la vera posta? Allo scacco
matto, sotto il piede del re sbalzato via dalla mano del vincitore, resta il
nulla, un quadrato nero o bianco. A forza di scorporare le sue conquiste
per ridurle all’essenza, Kublai era arrivato all’operazione estrema: la
conquista definitiva, di cui i multiformi tesori dell’impero non erano che
involucri illusori, s riduceva a un tassello di legno piallato (133).48
Allora Marco Polo parlò:- La tua scacchiera, sire, è un intarsio di due
legni…La quantità di cose che si potevano leggere in un pezzetto di legno
liscio e vuoto sommergeva a Kublai; già Polo era venuto a parlare dei
boschi d’ebano, delle zattere di tronchi che discendono i fiumi, degli
approdi, delle donne alle finestre…(133-4).
However, the duality of the frame goes further. Beyond being the story, which
encloses the stories, linking the timeless city spaces, the frame provides a means for
bridging the gap not just among cities, but also between the (reader’s) mind and (textual)
matter. The frame also acts like a mirror containing and reflecting a myriad of mirror
cities. Thus perceived, the book could be visualized as a hall of magic mirrors
‘suspended’, in both senses, by the frame.
that converge, projections, abstract forms, vectors of force. The other path goes through a space crammed
with objects ...” (“Exactitude” 74).
48 “…The
Great Khan tried to concentrate on the game: but now it was the game's reason that eluded him.
The end of every game is a gain or a loss: but of what? What were the real stakes? At checkmate, beneath
the foot of the king, knocked aside by the winner's hand, nothingness remains: a black square, or a white
one. By disembodying his conquests to reduce them to the essential, Kublai had arrived at the extreme
operation: the definite conquest, of which the empire's mulitiform treassures were only illusory envelopes:
it was reduced to a square of planed wood.”(131).
“Then Marco Polo spoke: "Your chessboard, sire, is inlaid with two woods…The quantity of things that
could he read in a little piece of smooth and empty wood overwhelmed Kublai; Polo was already talking
about ebony forests, about rafts laden with logs that come down the rivers, of docks, of women at the
windows...” (132).
41
I have deliberately insisted on choosing the term ‘suspend’ (rather than interrupt)
in this analysis of the frame’s function since it conveys both the idea of hanging (by
which support is provided yet allowing freedom of movement), and the idea of deferring
(rather than digressing). Suspension is a strategy for postponing, for pausing, the
development of the stories within the text (and consequently, their ending).
The purpose of this strategy is to keep the tempo of the narrative:
“... nel sapere incatenare una storia all’altra e nel sapersi interrompere al momento giusto:
due operazioni sulla continuità e la discontinuità del tempo. È un segreto di ritmo, una
cattura del tempo…nella narrazione in prosa per gli effetti che tengono, vivo il desiderio
d’ascoltare il seguito” (Lezioni 46).49
The frame gives a breathing space within the narrative. In fact, the protagonists
discuss the theme of space, real and imaginary:
KUBLAI- Non so quando hai avuto il tempo di visitare tutti I paesi che mi
descrivi. A me sembra che tu non ti sia mosso da questo giardino.
POLO:-Ogni cosa che vedo e faccio prende senso in uno spazio della
mente dove regna la stessa calma di qui, la stessa penonmbra,lo stesso
silenzio da fuscii di foglie (103-104).50
As the Emperor reflects on the cities, he finds the “space” between them, to be the most
precious aspect of the storytelling:“Ma ciò che rendeva piu prezioso a Kublai ogni fatto o
notizia…era lo spazio che restava loro intorno, un vuoto non riempito di parole. Le
49
“… knowing how to join one story to another, breaking off at just the right moment- two ways of
manipulating the continuity and discontinuity of time. It is a secret rhythm, a way of capturing time… in
prose narrative by those effects that make us eager to know what comes next.” (Six Memos 37-38)
50
“KUBLAI: I do not know when you have had time to visit all the countries you describe to me. It seems
to me you have never moved from this garden.
POLO: Everything I see and do assumes meaning in a mental space where the same calm reigns as here,
the same penumbra, the same silence streaked by the rustling of leave”(103).
42
descrizioni di città visitate da Marco Polo avevano questa dote: che ci si poteva girare in
mezzo col pensiero, perdercisi, fermarsi a prendere il fresco, o scappare via da corsa”
(39).51
The pleasure of lingering, conveyed by the two protagonists, (who are in the
magical garden of Kublai Khan, smoking pipes in their hammocks, playing games, telling
stories, dreaming) as if time were “a form of wealth to be spent at leisure and with
detachment”, is achieved by the technique of suspension combined with that of
repetition.52
Repetitions can be used to mark time (again, like the pendulum), to emphasize, to
create a rhythm, which with variation creates a flow. In Città, repetitions abound. The last
chapter of the frame closes and opens repeating the same words, allowing some
alterations, as variations in a musical theme.
At the opening of the chapter nine, the first words in the frame are: “Il Gran Kan
possiede un’atlante dove tutte le città dell’impero e del reami circonvicini sono disegnate
palazzo per palazzo e strada per strada, con le mura, i fiumi, i punti, i porti, le scogliere”
(137).53 Again, a few paragraphs later, just as we turn the page, the sentence begins: ”Il
Gran Kan possiede un atlante i cui disegni figurano la orbe terracqueo tutt’insieme e
continente per continente, i confini dei regni più lontani, le rote delle navi, i contorni
51
“But what enhanced for Kublai every event or piece of news reported by his inarticulate informer WaS
the space that remained around I, a void not filled with words. The descriptions of cities Marco Polo
visited had this virtue: you could wander through them in thought, become lost, stop and enjoy the cool air,
or run off”. (38)
52
Regarding narrative rhythm, Umberto Eco, in his Postille a “Il nome della rosa”, explains how novels
‘breath’ with different rhythms, some as whales, etc.
53
“The Great Khan’s owns an atlas where all the cities of the empire and the neighboring realms are drawn,
building by building, and street by street, with walls, rivers, bridges, harbors, cliffs.” (135)
43
delle coste, le mappe delle metropoli più illustri e dei porti più opulenti” (138).54 And
again, two paragraphs later, the atlas continues to be described: ”L’atlante raffigura anche
città di cui né Marco né i geografi sanno se ci sono o dove sono, ma non potevano
mancare tra le forme di città possibili…” (138).55
The iteration continues: “Il Gran Kan possiede un atlante in cui sono raccolte tutte
le mappe di tutte le città, quelle che elevano le loro mure su salde fondamenta, quelle che
caddero in rovina e furono inghiottite dalla sabbia, quelle che esisteranno un giorno e a
cui posto ancora non s’aprono che le tane delle lepre (139).56
And again, reappears in the closing part of frame, the book’s end: “L’atlante del Gran
Kan contiene anche le carte delle terre promesse visitate dal pensiero ma non ancora
scoperte o fondate….” (163). 57
Repetition, consequently, becomes another strategy, such as algorithms or recursive
methods, or embedding.
The frame of Città tells the story of Marco telling story after story In this case the
embedding encompasses only two levels. But this is an architectural strategy that creates
two parallel lines (of narrative development) at the same time: that of the frame and that
of the cities. At times, however the stories have other stories within. Such is the case, for
54
“The Great Khan owns an atlas whose drawings depict the terrestrial globe all at once and continent by
continent, the borders of the most distant realms, the ships’ routes, the coast lines, the maps of the most
illustrious metropolises and the most opulent ports.” (136)
55
“The atlas depicts cities which neither Marco nor the geographers know exist or where they are, though
they cannot be missing among the forms of possible cities … “ (137)
56
“The Great Khan owns an atlas in which are gathered the maps of all the cities: those whose walls rest on
solid foundations, those who fell in ruins and were swallowed up by the sand, those that will exist one day
and in whose place now only hares’ holes gape.” (137-8)
57 “The Great Khans atlas contains also the maps of the promised lands visited in thought but not yet discovered or founded … ” (134). 44
instance, of the five hidden cities, the “Città nascoste”, composed of a city within a city
within another city, endlessly; they demonstrate recursive structures. One example is the
last city, that of Berenice: “tutte le Berenici future sono già presenti in questo istante,
avvolte l’una dentro l’altra, strette pigiate indistricabili” (161).58 In literature, recursive
structures, allow both enclosure (even in never ending stories) and internal proliferations.
But the recursive technique acquires a unique ‘twist’, as it becomes a completely
mental puzzle, as the characters imagine that the other one imagines and so on: “Tutto
perché Marco potesse spiegare o immaginare di spiegare o riuscire finalmente a spiegare
a se stesso che quello che lui cercava era sempre qualcosa davanti a sé, e anche se si
trattava del passato era un passato che cambiava man mano egli avanzaba nel suo viaggio
…” (26).59
This embedding strategy makes the model of an infinite universe or, for that
matter, anything innumerable seem manageable. Embedding techniques, recursive
methods or algorithms, are used in mathematics not only to generate but primarily to
simplify the length or proliferation of a process; that is, to handle multiplicity. Due to the
elegant precision of mathematical formulas, such methods allow the text to be contained
in only a few pages. Aesthetically, mathematics searches for demonstrations, which are
considered the best since they are the most simple.
Narrative art as a combinatorial game evolves into an infinite system, which
contains other infinite systems. In view of this immense potential of the combinatorial
58
“ But what I wanted to warn you about is something else: all the future Berenices are already present in
this instant, wrapped one within the other, confined, crammed, inextricable” (163).
59
“All this so that Marco Polo could explain or imagine explaining or he imagined explaining or succeed
finally in explaining to himself that what he sought was always something lying ahead, and even if it was a
matter of the past it was a past that changed gradually as he advanced on his journey…” (28).
45
system, in the unfolding of the combinatorial process, worlds are multiplied. Simplicity
engenders complexity. Yet, this process is reversible. Through mathematical strategies,
the counterstatement is also true: complexity leads to simplicity. Calvino’s modular book,
takes full advantage of its malleability. If duality creates multiplicity, this multiplicity can
be not just “contained”, but also simplified or reversed.
Calvino is indeed aware of this reversibility: his techniques seem delightful games
that oscillate back and forth, always changing direction. Reversals are part of the game in
this system of doubles.
Incidentally, within the frame’s story, at that same moment, quoted above, where
Kublai Khan begins to “construct” his own cities, he claims that he will be the one to tell
the stories, expecting to reverse roles with Marco: “D’ora in avanti sarò io a descrivere le
città e tu verificherai se esistono e se sono come io le ho pensate” (43).60 In the central
chapter of the book, Kublai manages to tell his dream:
Ti racconterò cosa ho sognato stanotte, - dice a Marco…”
E Polo:- La città che hai sognato è Lalage...
-C’è qualcosa che tu non sai, - aggiunge il Khan. La luna ha datto alla
città di Lalage un privilegio più raro: crescere in leggerezza (73-74).
Subsequently, the five cities on this fifth chapter- Ottavia, Ersilia, Bauci, Leandra and
Melania- are all centered on the theme presented on the frame, by Kublai’s dream, that of
lightness (as will be examined on the next chapter.) 61
60
As the chapter is about to close, the Emperor insists. Having dreamt of a city, he wants to send Marco to
travel to that city and return, to confirm its veracity:”-Mettiti in viaggio, esplora tutte le coste e cerca questa
città, - dice il Kan a Marco. - Poi torna a dirmi se il mio sogno corrisponde al vero “ (55).
61
Within the frame, also, Kublai finds himself reflecting on the heaviness of his Empire and thus, begins to
dream of lighter cities (also Chapter 5).“È tempo che il mio impero già troppo cresciuto verso il fuori, pensava il Kan,-cominci a crescere al di dentro’…’È il suo stesso peso che sta scacchiando l’impero’, pensa
46
The skeletal frame of Città extricates itself, at least in part, from the playfully
entangled pattern delineated by the book’s index. As a simple embedding strategy, it tells
the story of a story or of many stories. But while the narrator is narrating, it also reflects,
meditates, about the tale that is being told. This cornice seems to spontaneously acquire,
a shape of its own, in a game where everything is calculated.
As mentioned previously, the frame functions as a mirror that includes other
mirrors, stories within a story. This embedding, its reflective nature, and this multiplicity
of mirrors, creates the illusion, among the protagonists, that it is difficult to know which
image is ‘real’ and which is ‘virtual’. Marco and Kublai are found making conjectures on
this aspect:
KUBLAI:- Neanch’io sono sicuro d’essere qui….
POLO: - Forse questo giardino esiste solo all’ombra delle nostre palpebre
abbasate…
KUBLAI: - Forse questo nostro dialogo si sta svolgendo tra due straccioni
sopranominati Kublai Kan e Marco Polo (103).62
POLO: - Forse del mondo è rimasto un terreno vago ricoperto da immondezzai, e
il giardino pensile della reggia del Gran Kan. Sono le nostre palpebre che li
separano, ma non si sa quale è dentro e quale è fuori (104).63
Again, the conversation is recaptured at the end of the chapter:
Kublai, e nei suoi sogni ora appaiono città legger come aquiloni, città traforate come pizzi, città trasparenti
come zanzariere, città nervatura di foglia, città linea della mano, città filigrana da vedere attraverso il loro
opaco e fittizio spessore.” (73)
62
“POLO: - I, too, am not sure I am here…
KUBLAI: - Perhaps this garden exists only in the shadow of our lowered eyelids…
KUBLAI: - Perhaps this dialogue of ours is taking place between two beggars nicknamed Kublai Khan
and Marco Polo…” (103-104).
63 “POLO:
Perhaps all that is left of the world is a wasteland, covered with rubbish heaps, and the
hanging garden of the Great Khan's palace. It is our eyelids that separate them, but we cannot know which
is inside and which outside” (104).
47
POLO: - …Forse questo giardino affacia le sue terraze solo sul lago della nostra
mente… (117).64
As the conjectures continue, what could have become a dizzying perspective ends with an
ironic twist.
KUBLAI: - Questa non mi pare una congettura che ci convenga…
POLO: - L’ipotesi è da escludere, allora. Dunque sarà vera l’altra: che ci siano
loro e non noi.
KUBLAI: - Abbiamo dimostrato che se noi ci fossimo, non ci saremmo.
POLO: - Eccoci qui, difatti (117).65
By self-reflection, the embedding, mirror-structure generates a book that reflects upon
itself. “A loop that allows a system to ‘perceive’ itself, to talk about itself, to become
‘self aware’” is by definition, a Gödel’ “strange loop”. 66 “Le città invisibili can be
considered a meta-text”, since it generates reflections on itself as well as the functioning
of narrative in general.
On the last chapter, the first part of the frame ends with this idea of infinite forms.
“Il catalogo delle forme è sterminato: finché ogni forma non avrà trovato la sua città,
nuove città continueranno a nascere” (140). (“The catalogue of forms is endless: until
every form has found its city, new forms will continue to be born”) (137). Hence, as each
form becomes a city, each city is a space looking for a form. The collection or catalogue
is potentially endless, due to its combinatorial nature.
64
“Perhaps the terraces of this garden overlook only the lake of our mind…” (117).
65 “KUBLAI:
To me this conjecture does not seem to suit our purposes. Without them we could never
remain here swaying, cocooned in our hammocks.
POLO: Then the hypothesis must be rejected. So the other hypothesis is true: they exist and we do not.
66
“Strange loops arise from a formal system …and it would not be too far to say that by virtue of having
such a loop, a formal system acquires a self”(D. Hofstadter 3).
48
The last sentence of the opening frame of chapter nine reveals the end of the atlas,
its last pages. “Nelle ultime carte dell’atlante si diluivano reticoli senza principio ne fine,
città a forma di Los Angeles, a forma di Kyoto- Osaka, senza forma” (140).67 The
paragraph continues, pointing towards the beginning of an end: “Dove le forme
esauriscono le loro variazioni e si disfano, comincia la fine delle città” (140).68 Right
before the end of the book, on its last pages, just as in the atlas, the message reappears:”Il
Gran Ka stava sfogliando nel suo atlante le carte delle città che minacciano negli incubi e
nelle maledizioni: Enoch, Babilonia, Yahoo, Butua, Brave New World” (163).69 Looking
at last the pages of his atlas, the Emperor claims everything is useless, ending in an evernarrowing spiral, at the infernal city.
In turn, Marco’ response, reveals how the perfect city could be constructed: pieceby-piece, of mixed fragments, separated intervals, “metterò assieme la città perfetta, fatta
di frammenti mescolata col resto”. The search, regardless of any discontinuity, must go
on: “Se ti dico che la città cui tende il mio viaggio è discontinua nello spazio e nel tempo,
ora più rasa ora più densa, tu non devi credere che si possa smettere di cercarla. Forse
mentre noi parliamo sta affiorando sparsa nei confini del tuo impero; puoi rintracciarla
ma in quel modo che ti ho detto”(163).70
Finally, Polo concludes, (with a phrase that is perhaps the most quoted by critics):
67
“In the last pages of the atlas there is an outpouring of networks without beginning or end, cities in the
shape of Los Angeles, in the shape of Kyoto-Osaka, without shape” (139).
68
“When the forms exhaust their variety and come apart, the end of cities begins” (139).
69
“Already the Kan was leafing through the atlas, over the maps of the cities that menace in nightmares and
maledictions: Enoch, Babylon, Yahoo, Butua, Brave New World” (164)).
70
“If I tell you that the city toward which my journey tends is discontinuous in space and time, now
scattered, now more condensed, you must not believe the search for it can stop. Perhaps while we speak, it
is rising, scattered, within the confines of your empire; you can hunt for it, but only in the way I have
said”(164).
49
L’inferno dei viventi non è qualcosa che sarà; se ce n’è uno è quello che è
già qui, l’inferno che abitiamo tutti I giorni, che formiamo stando insieme.
Due modi ci sono per non soffrirne. Il primo riesce facile a molti: accettare
l’inferno e diventarne parte fino al punto di non vederlo più. Il secondo è
rischioso e d esige attenzione e apprendimento continui: cercare e saper
riconoscere chi e cosa, in mezzo all’inferno non è inferno, e farlo durare, e
dargli spazio (164).71
Marco’s message gives a double option, but suggests that there is only one way out of the
inferno: “to seek and learn to recognize who and what, in the midst of inferno, are not
inferno, then make them endure, give them space” (165).
Significantly, the book closes with the word “space”. (This will be the central
theme of the next chapter in this research.)
71
“And Polo said: "The inferno of the living is not something that will be; if there is one, it is what is
already here, the inferno where we live every day, that we form by being together. There are two ways to
escape suffering it. The first is easy for many: accept the inferno and become such a part of it that you can
no longer see it. The second is risky and demands constant vigilance and apprehension: seek and learn to
recognize who and what, in the midst of the inferno, are not inferno, then make them endure, give them
space" (165).
50
Chapter Two: The Book as Space
“Questo avevo creduto di capire in quel mio lontano viaggio a Isfahan; che
la cosa più importante al mondo sono gli spazi vuoti…Il vuoto ha le sue
fantasie, i suoi giochi” (“Il mihrab” 221).
2.1 Invisible Spaces.
As the title states, this is a book about spaces, invisible spaces, worlds that cannot
be truly inhabited or even seen, yet potentially exist somewhere within its pages or in our
minds, “un paesaggio invisibile condiziona quello visibile” (Città 20) (“an invisible
landscape conditions the visible one”(Cities 20). Calvino’s imaginary cities suggest, from
its title, that only through the capacity to imagine – the ability to play with that, which is
invisible - can extraordinary things, like cities and stories, take place. Imagination turns
visible what is invisible.
For Calvino, literature is the search for that which is hidden, evasive, missing.
Precisely, according to Calvino, “the struggle of literature consists in an effort to escape
the confines of language, it stretches out from the utmost limits of what can be said; what
stirs literature is the call and attraction of what is not in the dictionary” (Uses of
Literature 18) (“La battaglia della letteratura è appunto uno sforzo per uscire fuori dai
confini del linguaggio; è dall’orlo estremo del dicibile che essa si protende; è il richiamo
da ciò che è fuori dal vocabolario che muove la letteratura”) (Una pietra sopra 211;
Saggi I: 217). Writing should be a way to transgress the space marked by language:
“Territori sicuri non esistono; l’opera è e dev’essere territorio di lotta”(“Per chi si
scrive?” 204). In Città, Marco Polo, the narrator, leads, even challenges his listener
Kublai Khan (and the reader) into possible ways to decipher his stories as sources of
signs in order to achieve his goals, to involve the reader into a game of “creative
51
reading”; that is, to make the reader an active participant in the game of creating spaces
in our minds.
Calvino wants to create a space where imagination can flourish – a dynamic shape
or form – to provoke the visualization of that which is invisible, to establish a game, a
challenge, a search for that which is absent. It is a space for the reader, for the reader’s
freedom, a space created by Marco’s tales of fantastic cities. “Quello che sta a cuore al
mio Marco Polo è scoprire le ragioni segrete che hanno portato gli uomini a vivere nelle
città, ragioni che potranno valere di là da tutte le crisi. Le città sono un insieme di tante
cose… Il mio libro s’apre e si chiude su immagini di città felici che continuamente
prendono forma e svaniscono, nascoste nelle città infelici” (Città ix).
The writer urges the reader to become aware of that, which is subtle, fleeting yet
essential for imagining a space that allows creative freedom. He wants a space, which
contaminates the air with spontaneity, where imaginations can breathe. As in the “The
Count of Monte Cristo”, the possibility of an exit, of escaping from the fortress of If,
freedom is to be found at that point where imagination does not coincide with reality. “Se
riuscirò col pensiero a costruire una fortezza da cui è impossibile uscire, questa fortezza
pensata o sarà uguale alla vera- e in questo caso è certo che di qui non fuggiremo mai [...]
- o sarà una fortezza dalla quale la fuga è ancora più dificileche di qui- e alllora è segno
ch qui una possibiltá…una possibilità di fuga esiste: basterà individuare il punto in cui la
fortezza pensata non coincide con la vera per trovarla” (RR II: 356).72
72
Italo Calvino was clearly influenced by Jorge Luis Borges. Adriano Piacentini, for instance comments:
“non si può impedire al pensiero di correre a Borges…ha avuto una profonda influenza su Calvino e su gli
ambienti da lui frequentati- e al “giardino dei pensieri che si biforcano…(Tra il cristallo e la fiamma 70).
Also, Calvino’s own comment, during Borges’ visit to Italy, is self revealing :“Comincerei col motivo
d’adesione più generale cioè l’aver conosciuto in Borges un’idea della letteratura come costruita e
governata dall’intelletto” (568).
52
The cities’ stories - Marco’s tales of fantastic cities - provide this space. In it the
reader can be free to initiate his own path, explore, focus his attention, to search for the
unexpected. Spontaneously, the book grants the reader a space in which to design his own
cities, a virtual space in which the reader’s gaze can rest at any point and be seduced to
traverse its subtle visual net. Reading becomes a chance to create to invent, to imagine,
designing within what already exists - the book, the net, the map, the city – other forms,
other exits, other games, other stories… This idea is not suddenly declared, but gradually
revealed. The reader is challenged by a game where there are no certainties. “The
connections between one element of the story and another were not always obvious to the
emperor…” (38).73 One must continue to search for that which seems to escape but at the
same time is waiting to be discovered, to be imagined, to inhabit our minds.
“Dimmi ancora un’altra città”, insisted Kublai Khan. Precisely this enthusiasm for
story telling is that enchantment which Calvino wants to cast on his reader. The intention
seems to share that secret which he finds in other great writers, “conservare intatta la
forza del desiderio” (“to preserve intact the force of desire”), in order to continue the art
of storytelling, like another Scheherazade74 (“Leggere, scrivere, tradurre, Saggi II: 1874).
From the visible world which can be blinding at times, the text provides mental
images which emerge as fleeting glimpses, discontinuous flashes granting privilege to
that which is invisible, potential. The end of the book emphasizes this view: “Alle volte
mi basta uno scorcio che si apre nel bel mezzo d’un paesaggio incongruo, un affiorare di
luci nella nebbia, il dialogo di due passanti che s’incontrano nel viavai, per pensare che
73
“Non sempre le connessioni tra un element e l’altro del raccontorisultavano evudenti all’imperatore…”
(39).
74
“Mondo scritto e mondo non scritto” 1874.
53
partendo di lì metterò assieme pezzo a pezzo la città perfetta, fatta di frammenti mescolati
col resto, d’istanti separati da intervalli, di segnali che uno manda e non sa chi li
raccoglie” (163).75
At the end of the fourth chapter, both Polo and Kublai describe their own idea of a
model. The Emperor’s model, leads to generalization and reduction. “– Eppure io ho
costruito nella mia mente un modello di città da cui dedurre tutte le città possibili, – disse
Kublai. – Esso racchiude tutto quello che risponde alla norma. Siccome le città che
esistono s’allontanano in vario grado dalla norma, mi basta prevedere le eccezioni alla
norma e calcolarne le combinazioni più probabili” (60).76
In contrast Marco’s model is based on exceptions and contradictions, on
differentiating.
Anch’io ho pensato un modello di città da cui deduco tutte le altre rispose Marco. – È una città fatta solo d’eccezioni, preclusioni,
contraddizioni, incongruenze, controsensi. Se una città così è quanto di più
improbabile, diminuendo il numero degli elementi abnormi si accrescono
le probabilità che la città ci sia veramente. Dunque basta che io sottragga
eccezioni al mio modello e in qualsiasi ordine proceda arriverò a trovarmi
davanti una delle città che pur sempre in via d’eccezione , esistono. Ma
non posso spingere la mia operazione oltre un certo limite: otterrei delle
città troppo verosimili per essere vere (69).77
75
“At times all I need is a brief glimpse, an opening in the midst of an incongruous landscape, a glint of
light in the fog, the dialogue of two passerby meeting in the crowd, and I think that, setting out from there, I
will put together, piece by piece, the perfect city made of fragments mixed with the rest, of instants
separated by intervals, of signals one sends out, not knowing who receives them.” (164)
76
"And yet I have constructed in my mind a model city from which all possible cities can be deduced,"
Kublai said. "It contains everything corresponding to the norm. Since the cities that exist diverge in varying
degree from the norm, I need only foresee the exceptions to the norm and calculate the most probable
combinations.” (69)
77
“I have also thought of a model city from which I deduce all the others,” Marco answered. “It is a city
made of exceptions, exclusions, incongruities, contradictions. If such a city is the most improbable, by
reducing the number of abnormal elements, we increase the probability that such a city exists. So I have
only to subtract exceptions from my model, and in whatever direction proceed, I will arrive at one of the
54
In the last section of the book, Marco insists on this discontinuity, one that must
be searched with persistence. “If I tell you that the city toward which my journey tends is
discontinuous in space and time, now scattered, now more condensed, you must not
believe the search for it can stop” (164). As discussed, this discontinuous character
allows space for fragments to be combined. This idea had already been suggested at the
introduction of the chapter, as a “calculation” whose “exactitude” is based on “residuals”:
“Io raccolgo le ceneri delle altre città possibili che scompaiono per farle posto e non
potranno più essere ricostruite né ricordate. Solo se riconoscerai il residuo d’infelicità che
nessuna pietra arriverà a risarcire, potrai computare l’esatto numero di carati cui il
diamante finale deve tendere, e non sbaglierai nel tuo progetto dall’inizio (60).78
In opposition to the opaque saturated, heavy world, Calvino presents cities
suspended above the void, intermingled or hidden under, above, beneath, which the
narrative net rescues through its subtle, thin threads. Its light, almost transparent
filigrana, its net structure allows infiltration through its open spaces. All in all, the net
sustains at a distance the view of other possible worlds, fleeting glimpses towards
potential, free spaces, new stories
2. 1. 1 Space as a Combinatorial Construction
Ma ciò che rendeva prezioso a Kublai ogni fatto o notizia riferita dal suo
inarticolato informatore era lo spazio che restava loro intorno, un vuoto
non riempito da parole. Le descrizioni di città visitate da Marco Polo
avevano questa dote: che ci si poteva girare in mezzo col pensiero,
cities, which, always as an exception, exist. But I cannot force my operation beyond a certain limit: I would
achieve cities too probable to be real” (69).
78
“I am collecting the ashes of the other possible cities that vanish to make room for it, cities that can
never be rebuilt or remembered. When you know at last the residue of unhappiness for which no precious
stone can compensate, you will be able to calculate the exact number of carats toward which that final
diamond must strive. Otherwise, your calculations will be mistaken from the very start” (60). 55
perdercisi, fermarsi a prendere il fresco, o scappare via da corsa (39).79
The most interesting aspect of Le città invisibili consists in the creation of
fictional space through ars combinatoria. The immensity of imaginary spaces, which
awaits the reader, testifies for the potential of Italo Calvino’s combinatorial game, that
consistent search for new forms which engages the reader in a game of combinatorial
freedom: “Ma la tensione della letteratura non è forse rivolta continuamente a uscire da
questo numero finito?” (Una pietra sopra 211; Saggi I: 217).80 (“But is the tension of
literature not continually striving to escape from this finite number?”Use of Literature
18). (Within mathematics, the branch, which explores limits, is that of infinitesimal
Calculus).
Italo Calvino’s concept of a book as a space will be the point of departure of this
part of the research. The conception of the book as a space can be viewed, in essence, as
a mathematical construction. Mathematical concepts are used as means, as tools, as
strategies to visualize, to give form, to design shapes, also, for that which is invisible.
They constitute a way to visually grasp the complexities of both the real and the
imaginary universe. Mathematics, as we will explore, also attempts to grasp that which is
fleeting, evasive yet possible to imagine.
79
“But what enhanced for Kublai Khan every event or piece of news reported by his inarticulate informer
was the space that remained around it, a void not filled with words. The descriptions of the cities Marco
Polo visited had this virtue: you could wander through then in thought, become lost, stop and enjoy the cool
air, or run off (39). In “Il viandante nella mappa”, we find the same idea in relation to a map’s empty
spaces: “ma sono proprio queste carte deserte, disabitate, che risvegliano nell’immaginazione il desiderio di
viverle dal di dentro, do rimpicciolirsi fino a trovare le propria via nel fitto dei segni, di percorrere, di
perdercisi” (Collezione di sabbia 27).
80
“But is the tension of literature not continually striving to escape from this finite number?” (Uses of
Literature 18).
56
From a mathematical perspective, the book space of Le città invisibili is
delineated by a combinatorial structure. In this book Calvino defines a space of imaginary
cities created by a combinatorial game. Mathematically, a space can be defined as a set of
points that follow specific rules or postulates, that is, some “constraints”, and, thus,
acquiring and delineating a structure. It is through the combinatorial process that these
constraints are established.
Conceptually, space is an unlimited expanse, a distance extending in all
directions, a continuous area that is free, available, unoccupied. Evidently, Calvino’s
‘book-space’, cannot be but bounded. In particular, it must be confined, as he states, by a
beginning and an ending81:“ … un libro (io credo) è qualcosa con un principio e una
fine… è uno spazio in cui il lettore deve entrare, girare, magari perdersi, ma a un certo
punto trovare un’uscita, o magari parecchie uscite, la possibilità d’aprirsi una strada per
venirne fuori” (Città vi).
Besides defining the book-space as being delimited in its opening and closure,
Calvino specifies that any book in order to be a book must have a construction, which
involves an itinerary. “Si deve poter scoprire un intreccio, un itinerario, una soluzione…”
(Città vi). According to Calvino, this book-space is one in which the reader can wander,
yet he adds it must have an itinerary, an intreccio, provide a solution, an exit or various
exit, “uno spazio in cui il lettore deve entrare, girare, magari perdersi, ma a un certo
punto trovare un’uscita, o magari parecchie uscite...” (Città vi). These concepts, which
81
Also, in “Il viandante nella mappa”, Calvino insists on the importance of following a route, from
beginning to end, drawing the analogy between traveling and narative structure: “Il seguire un percorso dal
principio alla fine dà una speciale soddisfazione sia nella vita che nella letteratura (il viaggio come struttura
narrativa)…” (Collezione di sabbia 22).
57
define the book’s construction, are fundamental to the mathematical analysis of the book
as a space.
The itinerary of Città is initially established by the index. An itinerary implies not
only space but also time. It relates time and space, pointing to the idea of travel, maps and
stories, all essential aspects of the space of the book, which will be analyzed in this
chapter 2. Departing from the index as an itinerary, reading the stories becomes traveling
through the book. Due to a combinatorial game, however, and its Oulipian character
many other itineraries could be traced, thus, creating a network.
The function of the index in Città is to allocate the cities within the space of the
book. Graphically, the combinatorial construction of the book generates a network in
which the cities are interlinked as they are mapped. Within the allocation of the cities in
the index each city is assigned a number and a theme, generating a combinatorial pattern,
which creates a mapping or a correspondence between two sets. Duality, central to the
book, engenders multiplicity. In this case, the two sets can be considered the theme and
the number assigned. This can also be viewed as a matrix. By creating a (binary)
correspondence between the cities and the book, the index becomes a map, as well as a
game, a guide for the reader to travel through the book. Following the trajectory
delineated in the index, we can graphically obtain either a zigzag pattern (connected by
straight lines) or a two dimensional spiral (using curved lines). This map of the cities
defines the architectural design of the book as a combinatorial structure. Through the
combinatorial itinerary of the index, mathematics provides not only a structure but also a
guide for the reader to travel through the book.
58
The structure of the book as a combinatorial game was explicitly designed and
exposed in a letter to Cesare Milanese on August 18th, 197482. For Calvino, this structure
is that of a “multifaceted network”. The image of the net conveys the interrelationships
between the nodes (cities) as well as the gaps or spaces between them. Once this structure
is established, not only the relationships or links become important but also the gaps, the
spaces in between. Without these the net would not be a net. These gaps not only allow
space but also are intrinsic parts of the net. Within the book, these spaces amidst spaces
82
Calvino’s letter reveals the design of his book:
A Cesare Milanese – Roma
Castiglione della Pescaia 18 agosto 1974
Caro Milanese,
Ho letto con gran piacere il tuo saggio pitagorico,* sul mio libro , o meglio sull’ indice del mio
libro . Costruendo quel sistema d’alternanza dei capitoletti delle varie serie, ho cercato di mettere in pratica
il sistema più semplice perché le serie non fossero tutte raggruppate e separate ma si allacciassero l’una
all’altra formando una continuità mossa e variata. Perciò ho stabilito l’ordine che rappresento nello schema
qui accluso. Le verticali sono le serie e le orizzontali i capitoli numerati normali di cinque capitoletti. Ma
siccome le prime orizzontali sono più corte, ho riunito in un capitolo introduttivo il “triangolo” iniziale e in
un capitolo conclusivo il triangolo finale, i quali naturalmente sono venuti più lunghi, di dieci capitoli
ciascuno.
Vedo che tu con un procedimento aritmeticamente più ingegnoso e complesso arrivi a ricostruire la mia
stessa figura… (Lettere 1250).
I
II
III
IV
V
VI
VII
VIII
IX
1
21
321
4321
54321
54321
54321
54321
54321
54321
54321
5432
543
54
5
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let the imagination pursue different paths, challenging constraints, allowing motion,
change, further combinations, even a way out.
In reference to these openings, Calvino comments in his Preface to Emilio
Cecchi’s book Mexico considering it an essential aspect of the artistic work.
È soprattutto una lezione di poetica che è certo la gemma del libro, nelle
riflessioni sui tessuti degli indiani Navajos: “Quando una donna Navajo
sta per finire uno di questi tessuti, essa lascia nella trama e nel disegno una
piccola frattura, una menda: “Affinché l’anima non le resti prigioniera
dentro al lavoro”. Questa mi sembra una profonda lezione d’arte: vietarsi
deliberatamente una perfezione troppo aritmetica e bloccata. Perché le
linee dell’opera, saldandosi invisibilmente sopra se stesse, costituirebbero
un labirinto senza via d’uscita, una cifra, un enigma di cui s’è persa la
chiave. Per primo, s’irretirebbe nell’inganno lo spirito che ha creato
l’inganno (“Cecchi in Messico”. Saggi I: 1044).83
Right from the beginning, Kublai Khan detects in Marco’s stories a “filigrana”. “Solo nei
resoconti d Marco Polo, Kublai Khan riusciva a discernere attraverso le muraglie e le
torri destinate a crollare, la filigrana d’un disegno cosi sottile da sfuggire al morso delle
termite.”
Filigree is a delicate and intricate ornamental work made from gold, silver or
other fine twisted wire or stitching of the same curving motifs. Moreover, it is an open
work, meaning that it is constructed so as to show openings through its substance.
Essential to this type of work are the “empty” spaces, just like in a net structure.
83
In “Lo scrittore e la città” (1960), Calvino, upon declaring Turin to be the ideal city for a writer,
comments: ”Non so come si faccia a scrivere in una di quelle città in cui immagini del presente sono così
soverchianti, così prepotenti, da non lasciare un margine di spazio e silenzio” Saggi II: 2708; Eremita a
Parigi 10. (“I do not understand how one could manage to write in one of those cities where images of the
present are so overwhelming and powerful that they leave the writer no margin of space and silence”
Hermit in Paris 6).
60
The concept of motion is fundamental in the combinatorial game. Spaces are
needed for mobility.84 They allow movement, a change of space through time and
location, like in a game of chess.
The index as a net is designed so as to create an “intelligent mechanism” which
triggers the formation of new spaces. The light, transparent structure of the net - that
which is incongruous, discontinuous- allows infiltration through its open spaces,
sustaining at a distance the view of other possible worlds, fleeting glimpses towards
unlived, free spaces, new stories.
As can be seen, mapping and combinatorial games are the two main aspects used
by Calvino in the construction of this book-space. Conceptually, these two mathematical
strategies are inherently connected. In fact, towards the closure of the book, the game (of
chess) and the map (Kublai’s atlas) become recurrently dominant, emphatically repeated
within the frame.
In simple terms, a map is a representation of a territory. Different maps help to
explore, to discover, to get to know a place.85 Yet maps are not fixed diagrams or
sketches, instead they are dynamic, always in transformation. Maps can be viewed as a
construction in constant metamorphosis; they belong to the “universe” of transformations
and intercommunications, just as the combinatorial game. They can be “comprehended
84
In “Il mihrab”, Calvino contemplates the possibility that a city constructed with a harmonious or (happy)
arrangement (disposition) of spaces, could in turn, lead to living it with happy disposition of spirit : “Forse
una città che è stata fatta seguendo una felice disposizione dei pieni e dei vuoti si presta a essere vissuta con
felice disposizione di spirito…” (Sabbia 222).
85
In regards to the theme of maps, the influence of Borges is again, undeniable. “Del rigor y la ciencia”
(“On Exactitude (Rigor) in Science”) constitutes a one paragraph story regarding the relationship between
maps and territories. Coincidentally, the story also develops on a concept in Lewis Carroll’s Sylvie and
Bruno Concluded where a map that had the scale of a mile to the mile. A character in Carroll’s last story
ironically points to the unpractical aspect of this fictional map: he claims that the country itself is used as a
map and is just as useful. Borges, in his story, imagines an empire where the science of cartography
becomes so exact that only a map on the same scale as the Empire should suffice.
61
“or approached through traveling, traversing in and out, by oscillating motions, by
perception and rationalization. On the other hand, combinatorial games assign different
places or locations, to a certain number of elements. This, by definition is in essence,
mapping.
The notion of the “literary work as a map” is considered by Calvino to be a “deep
rooted vocation in Italian literature”, (handed from Dante to Galileo): “Questa è la
vocazione profonda della letteratura italiana che passa da Dante a Galileo: l’opera
letteraria come mappa del mondo e dello scibile...” (Saggi I: 232-33).
Mapping consists in establishing a mathematical function, which defines a
relationship of correspondence, a link between two spaces, such as the earth and a globe,
or a book and its parts. Through mapping the elements of one space are assigned an
order, an arrangement; that is, they are allocated in reference to a second space. The first
space –the domain-, could be the model to the second –the range-, as we will see within
the cities, such as the cities and the sky, or the city of Venice, the implicit model within
Marco’s cities. (In fact, Venice can be viewed as an ever-changing map:“ Il grande centro
cartografico del Rinascimento è una città in cui il tema spaziale dominante è l’incertezza
e la variabilità, dato che i limiti fra terra e acqua cambiano continuamente; Venezia dove
le carte della Laguna cambiano continuamente”) (Collezioni di sabbia 26).
Mapping can then be viewed as a way to attribute meanings, by creating
correspondences and relationships among different objects. Through mapping meanings
are assigned and, as will be shown, mapping becomes a key strategy for the construction
of the book.
62
2.1.2
Combinatorial Game and the Unexpected
A combinatorial game is one in which at a certain point something unpredicted is
triggered, elicited. According to Gödel’s Incompleteness Theorem, in any formal system,
such as the combinatorial one, something always emerges that is not part of the system or
cannot be proved or explained by the system itself. There will always be a question,
which will not find an answer within the system. Any such system has the power or
strength to create “clandestine” elements. That is, “inadvertently but inexorably”, its
mapping is able to make a leap, “from inanimate constituents to animate patterns”
(Hofstadter 5).
Calvino commenting on Lucretius (De rerum natura) writes: “even while laying
down the rigorous mechanical laws that determine every event, he feels the need to allow
atoms to make unpredictable deviations from the straight line, thereby ensuring freedom
both to atoms and to human beings” (“Lightness” 9).86
Thus, intrinsic to the combinatorial game, which organizes the book, is the
component of the unforeseen. In Città, this element of surprise characterizes the cities,
becoming the essence of each city. At some point within the descriptions of the cities,
something turns around explicitly or implicitly, as the fundamental nature of the cities
appears before our eyes.
Initially, in Fillide, “at every point the city offers a surprise to your view” (90).
You enjoy a variety that can barely be grazed with your glance. However, as soon as you
realize that you must stay there forever, the city becomes invisible. As many other cities
86
According to Calvino, “Lucretius main concern is to prevent the weight of matter from crushing us.” In
fact, Lucretius insists on defining matter as being made of invisible parts that are “infinitely minute, light
and mobile” (“Lightness” 8-9).
63
in the book, Fillide becomes “a space in which routes are drawn between points
suspended in the void” (91). And here, the exceptional appears; that which occurs “by
surprise”: “Many are the cities like Phyllis, which elude the gaze of all, except the man
who catches them by surprise” (91).
In ogni suo punto la città offre sorprese alla vista… Felice chi ha ogni giorno
Fillide sotto gli occhi e non finisce mai di vedere le cose che contiene”, esclami,
col rimpianto di dover lasciare la città dopo averla solo sfiorata con lo sguardo. Ti
accade invece di fermarti a Fillide e passarvi il resto dei tuoi giorni. Presto la città
sbiadisce ai tuoi occhi, si cancellano i rosoni, le statue sulle mensole, le cupole…
segui linee a zigzag da una via all'altra, distingui zone di sole e zone d'ombra, …
Tutto il resto della città è invisibile. Fillide è uno spazio in cui si tracciano
percorsi tra punti sospesi nel vuoto, …. Milioni d'occhi s'alzano su finestre ponti
capperi ed è come scorressero su una pagina bianca. Molte sono le città come
Fillide che si sottraggono agli sguardi tranne che se le cogli di sorpresa (91-92).87
Calvino comments on yet another essential combinatorial aspect, that of”the
pleasure of puns and feeble jokes”. According to him “following the possibilities of
permutation and transformations implicit in language” attains the ludic effect (Uses of
Literature 20). This process, characteristically Oulipian,88 is based on the sudden
emergence of an unanticipated result: “the juxtaposition of concepts that we have
87
At every point the city offers surprises to your view…”Happy the man who has Phyllis before his eyes
each day and who never ceases seeing the things it contains” you cry, with regret at having to leave the city
when you can barely graze it with your glance. But it so happens that instead you must stay in Phyllis and
spend the rest of your days there. Soon the city fades before your eyes, the rose windows are expunged the
statues on the corbels the domes. Like all of Phyllis's inhabitants, you follow zigzaglines from one street to
another you distinguish the patches of sunlight from the patches of shade…All the rest of the city is
invisible. Phyllis is a space in which routes are drawn between points suspended in the void…Millions of
eyes look up at windows, bridges, capers, and they might be scanning a blank page. Many are the cities like
Phyllis, which elude the gaze of all, except the man who catches them by surprise” (90-91).
88
An essential interest to Oulipo was this surprisingly potential, playful, at times ludic, aspect of literature
when combined with mathematical concepts. Oulipo evolved from the Collège de Pataphysique, by Alfred
Jarry “the science of imaginary solutions” (W. Motte. Oulipo 1986, 197).
64
stumbled across by chance unexpectedly unleashes and a preconscious idea, an idea that
is half buried or erased […]” (Uses of Literature 21). 89
A similar development materializes with the fantastic 90 as explained in his essay
“Definizioni di territori: il fantastico”: ”Per i lettori d’Ariosto non si è mai posto il
problema di credere o di spiegare, per loro, come per i lettori del Naso di Gogol, di Alice
in Woderland, della Metamorfosi di Kafka, il piacere del fantastico si trova nello sviluppo
d’una logica le cui regole, il cui punto di partenza, o la cui soluzione, riservano delle
sorprese” (Pietra 261).91
Calvino’s definition of literature as combinatorial game states: “Literature is a
combinatorial game that pursues the possibilities implicit in its own material… but it is a
game that at a certain point is invested by an unexpected meaning” (Uses of Literature
22). It leads to exception.
The city of Armilla does not have anything a city should have: except for its water
pipes: “Se Armilla sia così perché incompiuta o perché demolita, se ci sia dietro un
incantesimo o solo un capriccio, io lo ignoro. Fatto sta che non ha muri, né soffitti, né
pavimenti: non ha nulla che la faccia sembrare una città, eccetto le tubature dell’acqua,
che salgono verticali dove dovrebbero esserci le case e si diramano dove dovrebbero
esserci i piani: una foresta di tubi che finiscono in rubinetti, docce, sifoni, troppopieno”
89
“È l’accostamento di concetti a cui si è pervenuti casualmente scatena inaspettatamente un’idea
preconscia, cioè a metà seppellita e cancellata dalla nostra coscienza, o anche soltanto allontanata, tenuta
in disparate, ma tale da poter affiorare alla coscienza se a suggerirla non è una nostra intenzione, ma un
processo oggettivo” (“Cibernetica e fantasmi”, Una pietra sopra 214). (This statement restates a key point
of surrealistic poetics, a point of interest for other correlated studies.)
90
“In Italian (as originally in French, I think) the words fantasia or fantastic…imply a detachment, a
levitation, the acceptance of a different logic based on objects and connections other than those of every
day life or the dominant literary conventions” (“Definitions of Territories: Fantasy” 72).
91
“Ariosto’s readers were never faced with the problem of believing or explaining. For them –as today for
the reader of Gogol’s “the Nose”, of Alice in Wonderland, or Kafka’s Metamorphoses – the pleasure of
fantasy lies in the unraveling of a logic with rules or points of departure or solutions that keep some
surprises up their sleeves” (The uses of Literature 72).
65
(48).92
Rising vertically and spreading out horizontally, they create “a forest of pipes”.
But the most intriguing aspect consists in the fact that the city is actually inhabited: “A
qualsiasi ora, alzando gli occhi tra le tubature, non è raro scorgere una o molte giovani
donne, snelle, non alte di statura, che si crogiolano nelle vasche da bagno, che si inarcano
sotto le docce sospese sul vuoto, che fanno abluzioni, o che s’asciugano, o che si
profumano, o che si pettinano i lunghi capelli allo specchio. Nel sole brillano i fili
d’acqua sventagliati dalle docce, i getti dei rubinetti, gli zampilli, gli schizzi, la schiuma
delle spugne” (48).93
Marco Polo playfully relates a logical fantastic explanation to this city’s mistery:
“The streams of water channeled in the pipes of Armilla have remained in the possession
of nymphs and naiads ... ” (“La spiegazione cui sono arrivato è questa: dei corsi d'acqua
incanalati nelle tubature d'Armilla sono rimaste padrone ninfe e naiadi…”) Since, as he
clarifies, they are used to moving around the alternate world of pipelines “they found it
easy to enter into the new aquatic realm, to burst from multiple fountains, to find new
mirrors, new games, new ways of enjoying the water” (49) (“è stato loro facile inoltrarsi
nel nuovo regno acquatico, sgorgare da fonti moltiplicate, trovare nuovi specchi, nuovi
92
“Whether Armilla is like this because it is unfinished or because it has been demolished, whether the
cause is some enchantment or only a whim, I do not know. The fact remains that it has no walls, no ceilings,
no floors: it has nothing that makes it seem a city, except the water pipes that rise vertically where the
houses should be and spread out horizontally where the floors should be: a forest of pipes that end in taps,
showers, spouts, overflows” (49).
93
“At any hour, raising your eyes among the pipes, you are likely to glimpse a young woman, or many
young women, slender, not tall of stature, luxuriating in the bathtubs or arching their backs under the
showers suspended in the void, washing or drying or perfuming themselves, or combing their long hair at a
mirror. In the sun, the threads of water fanning from the showers glisten, the jets of the taps, the spurts, the
splashes, the sponges' suds” (49).
66
giochi, nuovi modi di godere dell'acqua” (50). This joyful combinatorial creativity of the
nymphs concludes with their morning singing: a sign of their existence in this invisible
city. “Comunque, adesso sembrano contente, queste donnine: al mattino si sentono
cantare”. (“In any case, now they seem content, these maidens: in the morning you hear
them singing” 49-50).
In his essay, “Il richiamo dell’acqua” (RR IIII: 277-281), Calvino expresses the
idea that any city can be seen as a “liquid structure”, as a “net made of water threads”,
even as a “circumscribed” space, “constrained by vertical and horizontal lines of water”:
Il punto d’arrivo dell’acquedotto è sempre la città[...] Una città trasparente
scorre di continuo nello spessore compatto delle pietre e delle calci, una
rete di fili d’acqua fascia le mura e le vie. Le metafore superficiali
definiscono le città come agglomerato di pietra, ma ogni metropoli può
essere vista anche come una grande struttura liquida, uno spazio delimitato
da linee d’acqua verticali e orizzontali, una stratificazione di luoghi
soggetti a maree e inondazioni e risacche, dove il genere umano realizza
un ideale di vita anfibia che risponde alla sua vocazione profonda” (RR
III: 280-281).94
The combinatorial game displays its potentiality by the multiple shapes and forms it can
generate: the fifty-five cities are just a vivid example. All of the cities portray at a certain
point a reversal, a counterpoint in which something unique emerges. As said before
(2.1.1), from a mathematical standpoint, this aspect of Città invisibilli consists in the
creation of fictional space through ars combinatoria; the vastness of imaginary spaces
engages the reader of Città in the search for new forms.
Upon examination, this “unexpected meaning” of the combinatorial game
significantly becomes an integral part of the stories. In fact, this element becomes the
94
This passage echoes Venice, Marco’s implicit model in Città, as will be discussed.
67
essence of the story. Each story displays this trend: what begins as a description in a
certain way, following a specific trend of thought, suddenly follows a different direction.
Within the cities, each description appears to have a turning point opposing or contrasting
what has been said or seen up to that point establishes what is unique to that particular
city. Each city has its singolarità, a unique singular quality, which constitutes its special
character making it exceptional in some way.
Regarding, for instance, the city of Aglaura, Marco begins by telling us that there
is “little to tell”: “Poco saprei dirti d’Aglaura fuori delle cose che gli abitanti stessi della
città ripetono da sempre…” (“There is little I can tell you about Aglaura beyond the
things its own inhabitants have always repeated”). The possibility that nothing seems to
change is emphatically suggested: “Né l’Aglaura che si dice né l’Aglaura che si vede
sono forse molto cambiate da allora, ma ciò che era eccentrico è diventato usuale,
stranezza quello che passava per norma, e le virtù e i difetti hanno perso eccellenza o
disdoro in un concerto di virtù e difetti diversamente distribuiti….Se dunque volessi
descriverti Aglaura tenendomi a quanto ho visto e provato di persona, dovrei dirti che è
una città sbiadita, senza carattere, messa lì come vien viene” (68).95
However, none of these stories are true, yet they create a reliable account of the
city, whereas the random opinions, which might be inferred from living there, become
unsubstantial. Ironically, according to Marco Polo a reversal takes place: “In this sense
nothing that they say about the city is true…the city that they speak of has much of what
95
“Perhaps neither the Aglaura that is reported nor the Aglaura that is visible had greatly changed since
then, but what was bizarre has become usual, what seemed normal is now an oddity, and virtues and faults
have lost merit or dishonor in a code of virtues and faults differently distributed….So if I wished to
describe Aglaura to you, sticking to what I personally saw and experienced, I should have to tell you that it
is a colorless city, without character, planted there at random” (68).
68
is needed to exist, whereas the city that exists on its site exists less” (68). “In questo
senso nulla è vero di quanto si dice d’Aglaura, eppure se ne trae un’immagine solida e
compatta di città, mentre minor consistenza raggiungono gli sparsi giudizi che se ne
possono trarre a viverci. Il risultato è questo: la città che dice ha molto di quel che ci
vuole per esistere, mentre la città che esiste al suo posto, esiste meno” (67).
Thus, this city should be described “as a colorless city, without character, planted
there at random”. But, Marco turns the story around asserting, the singularity of
Aglaura.“Ma non sarebbe vero neanche questo: a certe ore, in certi scorci di strade, vedi
aprirtisi davanti il sospetto di qualcosa d’inconfondibile, di raro, magari di magnifico;
vorresti dire cos’è, ma tutto quello che s’è detto d’Aglaura finora imprigiona le parole e
t’obbliga a ridire anziché a dire”(67-68).96
Among the multiple double cities that will be examined, Marozia is made of two
cities at the same time: that of the rat and that of the swallow (154). Yet, the possibility
exists that “when you least expect it, you see a crack open and a different city appear”.
Just as well, we encounter an alternative: “that an instant later, it has already vanished”
(155).97
Perhaps everything lies in knowing what words to speak, which actions to
perform, and in what order and rhythm; or else, someone’s gaze, answer,
gesture is enough; it is enough for someone to do something for the sheer
pleasure of doing it, and for his pleasure to become the pleasure of others:
96
“But this would not be true, either: At certain hours, in certain places along the street, you see opening
before you something unmistakable, rare, and perhaps magnificent; you would like to say what it is, but
everything previously said of Aglaura imprisons your words and obliges you to repeat rather than say” (6768).
97
“Succede pure che, rasentando i compatti muri di Marozia, quando meno t’aspetti vedi aprirsi uno
spiraglio e apparire una città diversa, che dopo un istante è già sparita” (155).
69
at that moment, all spaces change, all heights, distances; the city is
transfigured, becomes crystalline, transparent as a dragonfly.
But everything must happen as if by chance, without attaching too much
importance to it, without insisting that you are performing a decisive
operation, remembering clearly that any moment the old Marozia will
return and solder its ceiling of store, cobwebs, and mold over all heads
(155).98
In “Il rovescio del sublime” Calvino’s description of the gardens of Kyoto approximates
the construction of his cities: “Tutto qui deve sembrare spontaneo e per questo tutto è
calcolato” (176).
Again, what seem to be of importance in the book are the exceptions to the rule.
This aspect is emphasized in more than one manner. For instance, that which escapes the
rules, the view, the net, appears to be the essence of the cities as well as the book. “At a
certain moment something in the mechanism is triggered, and literature gives birth to a
movement in the opposite direction, refusing to see things and say things the way they
had been seen and said until now …” (“Cybernetics and Ghosts” 23-24).
It is within these exceptions that the cities become unique. Ironically, this
exceptional character is what makes the cities most real: “una realtà che esplode in
fantasia” (Fiabe italiane, 53). Within these fantastic and remote cities we encounter
elements of our present everyday life; the exceptional is blended within the things of
daily living. Calvino concludes in his Presentation to his Fiabe, “Le fiabe sono vere” (1213). A similar conclusion could be drawn from his Città.
98
“Forse tutto sta a sapere quali parole pronunciare, quali gesti compiere, e in quale ordine e ritmo, oppure
basta lo sguardo la risposta il cenno di qualcuno, basta che qualcuno faccia qualcosa per il solo piacere di
farla, e perché il suo piacere diventi piacere altrui: in quel momento tutti gli spazi cambiano, le altezze, le
distanze, la città si trasfigura, diventa cristallina, trasparente come una libellula. Ma bisogna che tutto capiti
come per caso, senza dargli troppa importanza, senza la pretesa di star compiendo una operazione decisiva,
tenendo ben presente che da un momento all’altro la Marozia di prima tornerà a saldare il suo soffitto di
pietra ragnatele e muffa sulle teste”(155).
70
The city of Leonia, for instance, reflects an urgent urban and ecological problem,
that of garbage. In Leonia “every morning they wash with just –unwrapped cakes of soap,
wear model refrigerators, still unopened tins listening to update radio” (114). But the
story turns into the opposite direction. What the inhabitants seem to enjoy, according to
Marco’s wondering is, instead, “the joy of expelling, discarding, cleansing itself of a
recurrent impurity” (114). “La città di Leonia rifà se stessa tutti i giorni: ogni mattina la
popolazione si risveglia tra lenzuola fresche, si lava con saponette appena sgusciate
dall’involucro, indossa vestaglie nuove fiammanti, estrae dal più perfezionato frigorifero
barattoli di latte ancora intonsi, ascoltando le ultime filastrocche dall’ultimo modello
d’apparecchio” (113).99
Its garbage defines the form of the city, its ever-growing perimeter:“l'imponenza
del gettito aumenta e le cataste s'innalzano, si stratificano, si dispiegano su un perimetro
più vasto. …. rinnovandosi ogni giorno la città conserva tutta se stessa nella sola forma
definitiva: quella delle spazzature d'ieri che s'ammucchiano sulle spazzature dell'altro ieri
e di tutti i suoi giorni e anni” (114).100
Calvino’s humorous turning point makes this city unforgettably real. “The fact is
that street cleaners are welcomed like angels, and their task of removing the residue of
yesterday's existence is surrounded by a respectful silence, like a ritual that inspires
devotion, perhaps only because once things have been cast off nobody wants to have to
99
In “La poubelle agrée”, Calvino describes writing as the art of throwing away.
100
“The bulk of the outfow increases and the piles rise higher, become stratified, extend over a wider
perimeter…As the city is renewed each day, it preserves all of itself in its only definitive form: yesterday's
sweepings piled up on the sweepings of the day before yesterday and of all its days and years and decades”
(115).
71
think about them further” (114).101 In this endlessly increasing perimeter, “nobody
wonders where, each day, they carry their load of refuse. Outside the city, surely...”
(114).102
The unexpected, that element of surprise, which derives from the combinatorial
game, is the main distinction that is of relevance to this study. Among all those
“mappings” involved in the game, there are always some that escape the “system”, that
seem to challenge the rules, that emerge as something unique, singular. In his essay,
“L’orecchio, il cacciatore il pettegolo”, Calvino praises Galileo, precisely on this aspect:
“l’osservatore delle macchie del sole e della luna, del moto dei pianetti, il ragionatore che
non si faceva scrupolo d’accumulare prove per ridurre la terra a rango di pianeta in
mezzo agli altri, quale obiettivo poneva alla scienza se non il render conto della
singolarità contro ciò che pretendeva essere la norma…” (Enciclopedia103).
This is the essence of Calvino’s combinatorial art, this book and each one of the
cities.
2.1.3 The Book and the City
“La metafora più famosa nell’opera di Galileo – e che rachiude in sé il
nocciolo della nuova filosofia – è quella del libro della natura scritto in
linguaggio matematico” (“Il libro della natura in Galileo” 853).
The city is a recurrent theme in Calvino’s work, both in his fiction and his essays
where he show his interest not just in the city itself but also on how this subject has been
101
“Certo è che gli spazzaturai sono accolti come angeli, e il loro compito di rimuovere i resti dell’esistenza
di ieri è circondato d’un rispetto silenzioso, come un rito che ispira devozione, o forse solo perché una volta
buttata via la roba nessuno vuole più averci da pensare.” (113)
102
“Dove portino ogni giorno il loro carico gli spazzaturai nessuno se lo chiede: fuori della città,
certo…”(113-114)
72
addressed by other writers. Calvino perceives an analogy rooted deep into the novel: the
correlation between the book and the city.
The metaphor between the city and the book, the world and the book is a topic in
various essays. In “Natura e storia del romanzo”(1958) Calvino praises Balzac’s
perspective of the city, as the discovery of a new place in constant metamorphosis:
“Anche in Balzac: pur tutto calato com’è nella scoperta del grande nuovo continente che
gli s’apriva davanti, la città, l’infinita Parigi, i continui mutamenti di fortuna d’una
società in movimento. Balzac infatti è colui che scopre la vitalità naturale quasi biologica
della città (Saggi I: 34-35).
According to Calvino, the city becomes a novel for Balzac: “[…] far diventare
romanzo una città”. The term “città-romanzo” (city-novel), appears in Calvino’s
introduction of Balzac’s Ferragus (1973) where the city, as protagonist, is portrayed as a
living being: “[…] far si che in ogni mutevole momento la vera protagonista sia la città
vivente...questa è l’impresa cui Balzac nel momento in cui comincia a scrivere Ferragus
si sente chiamato” 103 (“La città romanzo in Balzac” I: 775).
In “Vento in una città”, Calvino’s description of Turin also ends in this analogy
between the city and a book as the wind begins to blow in this “windless city”.104 “Torino
è una città senza vento. Le vie sono canali d’aria ferma che si perdono all’infinito come
un urlo di sirena… Quando il vento nasce nella città e si propaga di quartiere in quartiere
103
“To work in such a way that at every changing moment the true protagonist was the living city…this is
what Balzac impelled to when he began to write Ferragus.”(“The city as a protagonist in Balzac” Uses
182).
104
In 1945, Calvino settles in Turin where he studies at the University, graduating in 1947 with a thesis on
Joseph Conrad.
73
in lingue di un incendio incolore, la città s’apre ai miei occhi come un libro…”(RR III:
952)
The city is a book, a legible text to be read and be written. Paris is for Calvino the
readable city. In 1967, Calvino moves to Paris where he lived on and off for fifteen years.
There, in 1973, a year after finishing Le città invisibili (1972), he became a foreign
member of Oulipo. Yet, he comments, “this city has never appeared in the things I write”
(Hermit in Paris 167). (Analogously, Venice, in Città, remains implicit.) He emphasizes,
“Maybe to write about Paris I ought to leave, to distance myself from it, if it is true that
all writing starts from a lack or an absence…” (167). Thus, echoes the central city of
Bauci in Città; let the city become “an inner landscape for the imagination to start to
inhabit that place” (167). “Dirò meglio: bisogna che un luogo diventi un paesaggio
interiore, perché l’immaginazione prenda ad abitare quell luogo… (Eremita 171); a
statement which describes the process of reading Città invisibili.
Cities turn for him into one single city “a single endless city where the differences
which once characterized each of them are disappearing. He states this idea runs through
his book Città invisibili, an idea that came “from the way many of us now live: we
continually move from one airport to another, to enjoy a life that is almost identical no
matter what city you find yourself in”(169). In Le città invisibili, on Chapter VIII, (“Città
continue 2”) the description of the city of Trude depicts this idea, which is central to the
“Continuous cities”:
Se toccando terra a Trude non avessi letto il nome della città scritto a
grandi lettere, avrei creduto d’essere arrivato allo stesso aeroporto da cui
ero partito. I sobborghi che mi fecero attraversare non erano diversi da
quegli altri, con le stesse case gialline e verdoline. Seguendo le stesse
frecce si girava le stesse aiole delle stesse piazze. Le vie del centro
mettevano in mostra mercanzie imballaggi insegne che non cambiavano in
74
nulla. Era la prima volta che venivo a Trude, ma conoscevo già l’albergo
in cui mi capitò di scendere; avevo già sentito e detto i miei dialoghi con
compratori e venditori di ferraglia; altre giornate uguali a quella erano
finite guardando attraverso gli stessi bicchieri gli stessi ombelichi che
ondeggiavano.
Perché venire a Trude? mi chiedevo. E già volevo ripartire.
– Puoi riprendere il volo quando vuoi, – mi dissero, – ma arriverai a
un’altra Trude, uguale punto per punto, il mondo è ricoperto da un’unica
Trude che non comincia e non finisce, cambia solo il nome all’aeroporto
(128).105
Journeys, whether short or international, are no longer explorations of different
places, but “simply a movement from one point to another between which there is an
empty interval, a discontinuity…” (Hermit 169). But they also establish moving
relationships: perhaps to find these, the author writes from one invisible and anonymous
point (171).
Paris becomes for Calvino “a giant reference work, a city which you can consult
like an encyclopedia: whatever page you open gives you a complex list of information
that is richer than that offered by any other city” (Hermit in Pars 171-172). But “in the
same way that we can ‘read’ the city as a reference book… we can interpret Paris… as a
book of dreams” (173). Paris is seen by Calvino “like a huge lost-property office, a little
like the moon in Orlando Furioso which gathers up everything that has been lost in the
105
“If on arriving at Trude I had not read the city's name written in big letters, I would have thought I was
landing at the same airport from which I had taken off. The suburbs they drove me through were no
different from the others, with the same little greenish and yellowish houses. Following the same signs we
swung around the same flowerbeds in the same squares. The downtown streets displayed goods, packages,
signs that had not changed at all. This was the first time I had come to Trude, but I already knew the hotel
where I happened to be lodged; I had already heard and spoken my dialogues with the buyers and sellers of
hardware; I had ended other days identically, looking through the same goblets at the same swaying navels.
Why come to Trude I asked myself. And I already wanted to leave.”
"You can resume your flight whenever you like”, they said to me, "but you will arrive at another Trude,
absolutely the same, detail by detail. The world is covered by a sole Trude, which does not begin and does
not end. Only the name of the airport changes” (128).
75
world… “This city invites you to make a collection of everything, because it accumulates
and classifies and redistributes…” (174).106
As Calvino stated, in Città the symbol of the city allowed him to “concentrate all
his reflections, experiences and conjectures.” In this convergence between the book and
the city he seems to look at the city as if it were the pages from a book where words
accumulate as traces, evidence of the life that inhabits them. On Città’s first chapter, the
city of Zaira (Città e memoria 3), contains its past written within the city: ”Inutilmente,
magnanimo Kublai, tenterò di descriverti la città di Zaira…la città non dice il suo
passato, lo contiene come le linee d’una mano, scritto negli spigoli delle vie, nelle griglie
delle finestre, negli scorrimano delle scale, nelle antenne dei parafulmini, nelle aste delle
bandiere, ogni segmento a sua volta di graffi, seghettature, intagli, svirgole (Città, 11). 107
Furthermore, in Città the city is explicitly compared to a written page. “Lo
sguardo percorre le vie come pagine scritte: la città dice tutto quello che devi pensare, ti
fa ripetere il suo discorso e mentre credi di visitare Tamara non fai che registrare i nomi
con cui essa definisce se stessa e tutte le sue parti” (“La città e i segni 1” 14).108 Yet it is
also plainly portrayed as a blank page, as a challenge to continue searching. “Milioni
106
The texts from “Hermit in Paris” are from an interview with Valerio Riva for Swiss Italian TV in 1974.
They were published in the same year in a limited edition, in Lugarno by Edizioni Pantarei with four
drawings by Giuseppe Ajmone (Author’s note).
107
“The city however does not tell of its past, but contains it like the lines of a hand, written in the corners
of the streets, the gratings of the windows, the banisters of the steps, the antennae of the lightning rods, the
poles of the flags, every segment marked in turn with scratches indentations, scrolls” (10-11).
108
“Your gaze scans the streets as if they were written pages: the city says everything you must think,
makes you repeat her discourse, and while you believe you are visiting Tamara you are only recording the
names with which she defines herself and all her parts.”(15)
76
d’occhi s’alzano su finestre ponti capperi ed è come scorresero una pagina bianca” (92).
109
Calvino could be seen as the traveler of cities, the explorer of city spaces. He is a
viewer, a reader and a narrator of this city space. Thus, the idea of the city converges for
Calvino with the problem of writing. The relationship between the city and writing
focuses his attention :“È la presenza della scrittura, la potenzialità del suo uso continuo
che la città deve trasmettere…la città ideale è quella su cui aleggia un pulviscolo di
scrittura che non si sedimenta né si calcifica” (“La città: epigrafi e graffiti” Sabbia 108).
There is a persistent presence in his writings of an intense and constant concern for the
city as space and its visual representation (“Città pensata: la misura degli spazi” 111).
“The narrative voice enters the city defining…its layout and architectural features, and
then probes into the urban space from the inside, interrogating its dimensions, volumes,
voids, immobility, melancholy, lights and shadows. What emerges is the narrating
observer’s vigilant desire to comprehend the city, to grasp the uniqueness of its soul…”
(Modena 10).110
In 1969, three years prior to publishing his book, Calvino proposes a study of the
city as a symbol, an alternative to the topic of his article on literature as the projection of
desire: “Per esempio resta aperta la via per uno studio del simbolo città dalla rivoluzione
industriale in poi, come proiezione dei territori e dei desideri dell’uomo contemporaneo”
109
“Millions of eyes look up at windows, bridges, capers, and they might be scanning a blank page” (91).
110
I. Calvino. “L’arcipelago dei luoghi immaginari”.1981 Sabbia: 545-549; “Il silenzio e la città: per Fabio
Borbottoni” 1982 Romanzi e racconti III: 779-890. “Viaggio nelle città di de Chirico.” Romanzi e racconti
III: 397-406.
77
(“La letteratura come proiezione del desiderio: per l’Anatomia della critica di Northrop
Frye” 245).111
For Calvino, desire is an essential part in literary imagination playing a
fundamental role in this ‘visualization’ of the ‘invisible cities’, where these inner cities of
the imagination become highly visual, where one can move. In this imaginative
projection of desire, there seems to be a search for a utopia, a utopian city. As Calvino
states in his presentation to Invisible Cities, even if we cannot find the utopian city we
cannot stop looking for it: “…città d’utopia (che anche se non scorgiamo non possiamo
smettere di cercare”(Città x).112 In the third essay of his trilogy devoted to Fourier,
Calvino insists:
In a word, utopia is not as a city that can be founded by us, but that can
found itself in us, build itself… in our ability to imagine it, to think it out,
to the ultimate degree, a city different from, a city that claims to inhabit
us, not to be inhabited, thus making us possible inhabitants of a third city,
different from all the habitable or inhabitable cities of today; a city born of
the mutual impact of new conditionings both inner and outer“ (“On
Fourier III: A Utopia of Fine Dust” 252). 113
His essay concludes with an image of utopia, which coincides, with Marco’s
discontinuous model of an ideal city, that which the reader is invited to search for in
Città: “something else which must be sought in the folds, in the shadowy places, in the
countless involuntary effects that the most calculated system creates without being aware
111
This consists in a critical review of Northrop Frye’s Anatomy of Criticism.
112
Similarly, at the end of Città, Marco Polo warns Kublai Khan : “Se ti dico che la città a cui tende il mio
viaggio è discontinua nello spazio en el tempo, ora più rada ora più densa, tu non devi credere che si possa
smettere di cercarla” (163).
113
“Insomma l’utopia come città che non potrà essere fondata da noi ma fondare se stessa dentro di noi,
costruirsi pezzo per pezzo nella nostra capacità d’immaginarla, di pensarla fino in fondo, città che pretende
d’abita re noi non d’essere abitata , e così fare di noi I possibili abitanti d’una terza città, diversa dall’utopia
e diversa da tutte le città bene o male abitabili oggi, nata dall’urto tra nuovi condizionamento interiori ed
esteriori.” (Pietra 306).
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that perhaps the truth lies right there” (The uses of Literature 255): “Oggi l’utopia che
cerco è più solida di quanto non sia gassosa: è un’utopia polverizzata, corpuscolare,
sospesa” (Una pietra sopra 308).114
Again, in his presentation of Invisible Cities, he claims he wrote Città as a last love poem
to a city in crisis: “Che cosa è oggi la città per noi? Penso d’aver scritto qualcosa come un
ultimo poema d’amore alle città nel momento in cui diventa sempre più difficile viverle
come città. Forse stiamo avvicinandoci a un momento di crisis della vita urbana e Le città
invisibili sono un sogno che nasce dal cuore delle città invisibili” (Città x).
While writing his book, he was working on a magazine (Ali Baba), which never
came out. However, many of his ideas for his book can be found in “Lo sguardo
dell’archeologo” (Una Pietra sopra 318-321). “Calvino’s text is indebted to the Alì Babà
circle’s oral and written dialogues because the visual and conceptual emblem of their
prospective magazine was the city, and because Ginzburg, Celati, and Gabellone put
forth a cognitive and epistemological model rooted in fragmentation and alterity (i.e., the
diverse, the otherwise, the nonlinear and peripheral)” (Modena 15).
On yet another article, “Gli dei della città”, Calvino comments: “È con occhi
nuovi che oggi si pone a guardare la città, e ci si trova davanti agli occhi una città
diversa…sono elementi che si compongono in una mappa intricata e fluida, difficile a
ricondurre all’essenzialità d’uno schema. Ma è qui che bisogna partire per capire- primocome la città è fatta, e scondo come la si può fare” (Una pietra sopra 343).
Accordingly, the intention is to explore these “essential elements”, through the
fundamentally mathematical aspect of mapping, both “intricate” as the net and “fluid”, as
the narrative game. How is the city made and how can it be constructed?
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This book-space designed by combinatorial art can be viewed as an architectural
construction. The image of the city and the network structure of the book constitute two
main elements for the design of Città. They are so intricately linked that they seem
indistinguishable. (Coincidentally, in mathematical logic, when an element A implies a
second element B, and element B implies A, a relationship of equality is established
between the elements.)
The mathematical strategy engaged is that of mapping: creating a correspondence
(between the city and the book). In fact, an analogy is the mathematical equivalent to
mapping. Mathematically, a one to one correspondence between the elements of two sets
that preserves the structural properties of the domain can be denominated as an
isomorphism. Conceptually, this constitutes mapping.
In Città, a correspondence between two sets is established between the city and
the book: one points to the other, both explicitly and implicitly, creating a book-city or
city-book. That is, through a bi-directional mapping, the book and the city seem to
become one.
Besides the mapping or plotting of the different cities, the combinatorial game
draws relationships among these “points” which are the cities. Beyond creating an
itinerary, there is the possibility of multiple itineraries, that is, a network, relating the
various cities within this space. The image of the city and its variations creates a sense of
unity, of coherence within these fifty-five cities. In this mapping, the correspondence is
such that an intricate, subtle pattern approximates the two elements involved: the book
and the city.
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2. 2 Traveling Through the Cities
“Quegli altri vagabondi che chiamiamo matematici.”
(Confessioni, S. Agostino)
In this city/book – book/city, traveling through the invisible cities means reading
the book. The book space becomes a traveling space, a story space. This reversibility
could also be considered as part of the book’s duplicity.
As stated before in Calvino’s words, “a book is a space with a beginning and an
end”. Traveling from one point to another means to narrate. In travels as in narrative, the
dimension of time must be added to that of space. Moreover, established by these
variables of space and time, maps are intrinsically related to both traveling and story
telling. In the story space of the book/city, the reader becomes a traveler, an explorer.
The relation between maps and travels goes in both directions, one points to the
other. In “Il viandante nella mappa”, Calvino states that the first need to fix a place on
paper is linked to travel (“il primo bisogno di fissare sulla carta i luoghi è legato al
viaggio” Sabbia 21). According to Calvino, it is necessary to comprehend that our image
of the dimension of time together with that of space is at the origin of cartography.115
Having established this correlation, “The geographic map even if static presupposes a
narrative idea, it is conceived in relation to an itinerary; it is an Odyssey.” (“La carta
geografica insomma, anche se statica, presuppone un’idea narrativa, è concepita in
funzione d’un itinerario, è Odissea” (Sabbia 23).
For centuries Marco Polo’s travels, recorded in his book Il Milione, has been a
great source of inspiration not just for explorers, such as Christopher Columbus, but also
115
“La necessità di comprendere in un’immagine la dimensione del tempo insieme a quella dello spazio è
all’origine dell cartografia” (Sabbia: 22).
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for cartographers and writers. In Venice, 1459, Fra Mauro’s world map, inspired on
Polo’s travels, appears upside down, inverted, with the south on top, in contrast to
Ptolemy’s concept, which Fra Mauro regarded as insufficient for many parts of the world.
“Vediamo l’Europa diventare piccola in confronto al resto del mondo nella carta di Fra
Mauro (1459) uno dei primi planisferi disegnato in base ai resoconti di Marco Polo e
delle circumnavigazioni dell’Africa, e cui l’inversione dei punti cardinali accentua il
capovolgimento di prospettive” (Sabbia 25).116
According to Roberto Amalgià, this map was considered “the greatest memorial of
medieval cartography”.117
Traveling engenders new ideas: stories generate maps and maps that inspire
stories, which, in turn propel new travels; this sequence may happen in any order. In his
Presentation of Città, Calvino refers to Il Milione not just as a source of inspiration for its
fantastic and exotic scenography,” in tutti i secoli ci sono stati poeti e scrittori che si sono
ispirati al Milione come una scenografia fantastica ed esotica”, but also as an “imaginary
continent” where other literary works “find their space”: “Solo Le Mille e una notte
possono vantare una sorte simile: libri che diventano come continenti immaginari in cui
altre opere letterarie troveranno il loro spazio; continenti dell’“altrove” oggi che
l’“altrove” si può dire che non esista più, e tutto il mondo tende a uniformarsi” (Città
viii).
116
Fra Mauro’s map is inversely oriented with the south in the upper part of the map. This was usual in
Muslim maps, contrary to Ptolemy’s map where the north was placed in the superior (upper) part.
117
Monumenta cartographica vaticana. (Rome 1944) I: 32–40.
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The Book of Marvels of the World relates Polo’s travels practically five hundred
years prior to Calvino’s book. Here, the aim is to use Calvino’s text (a rewriting of Il
Milione) as a “map” for our imagination to travel through the book or through the cities.
This book space is essentially a mathematical construction. Many mathematical
concepts, ideas or terms stand out explicitly to depict the cities. They serve as visual
designs, constructs or tools for our imagination. Mathematics renders visible what is
invisible. Mathematics, like the cities, may be invisible; yet, they can be imagined as they
attempt to approximate reality. Calvino’s cities, like mathematics, even though invisible,
can become imaginable. They end up not being as fantastic as they resemble those
“spaces” we live in.
As said, to read the book equals traveling through the cities (Even, the city of
Diomira, is divided in nine sections or quartiere just as the book, or “book-city”,
comprises nine chapters.) This travel inspires mathematical reflections that one would
most likely not think of without reading the book.
2. 2. 1 Mathematical Constructions
In this part of my research, I intend to explore and demonstrate the relevance of
mathematics in the construction of the book as a “space”, as defined by Calvino.
Calvino’s interest in mathematics can be initially perceived from the index of the
book. The index ‘ combinatorial nature combines the concepts of mapping and the
concept of game, two facets that, both open and close the book, as in a spiral. (The spiral
figure will be discussed on the next part of this research.)
His involvement with Oulipo, the aforementioned group of mathematicians and
writers, is perhaps of most importance in his contact with the world of mathematics. As
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mentioned, he considered Città in particular, to be a very “oulipian” book, due above all
to its index (“molto oulipiano soprattutto per il suo indice”).118
Posthumously, Numbers in the Dark (1993) was published, a collection including
the story “The Burning of the Abandoned House”. In Esther Calvino’s introduction to
Numbers in the Dark, she explains that: “The Burning of the Abandoned House” was the
result of a somewhat vague request from IBM: how far was it possible to write a story
using the computer? This was 1973 in Paris when it wasn’t easy to gain access to data
processing equipment. Undaunted, Calvino gave the project a great deal of his time,
carrying out by himself all the operations the computer was supposed to do.“(2)
Esther Calvino (his wife) explains that he had planned this story to be published by
Oulipo “as an example of ars combinatoria and a challenge to his own mathematical
abilities. (Italo Calvino suggested the title of the Oulipian collection Atlas of Potential
Literature.)
Calvino also came from a family of scientists. His father Mario was an
agronomist; his brother Floriano was an internationally known geologist; and his mother
Eva Mameli Calvino, a botanist (1886-1979). Furthermore, his mother not only had a
degree in Mathematics (Cagliari) and a second degree in Natural Science (Pavia), but at
the age of twenty-nine, was the first woman in Italy to become a university professor.
(Dell’Arti 56).
118
Besides those already mentioned (I, 4), he also wrote other Oulipian works, His books Il castello dei
destini incrociati (1969) and Se una notte d’inverno un viaggiatore (1979) are considered “Oulipian” works.
Interestingly, Le città invisibili (1972) was published in the period between these two books. Regarding Se
una note … the author also wrote, “How I wrote one of my books”, included in Oulipo Laboratory. In this
essay, the author delineated the structure of the book as well as the Oulipian devices used.
84
He was known to be an avid reader of Scientific American, a fact that, besides
showing his interest, kept him informed on the latest scientific and mathematical
research, discoveries and inventions. His essays, letters119 and stories testified his interest
in mathematical sciences, both in the history and the developments of these fields.
Even if we may not know exactly the extent of his mathematical knowledge, his
work, in this case, Città invisibili evinces a rather keen mathematical agility: Calvino
seems comfortable “playing” with mathematical “games”, transforming mathematical
patterns and concepts into stories and structures. Whether intentional or not, explicit or
implicit, reading the city descriptions in Città invisibili, provokes mental images that can
be viewed as essentially mathematical.
Mathematics is neither the study nor the science of numbers: it is the science of
structures, patterns and relationships. Mathematics examines structures of forms or
movement, which can be real or imaginary, visible or in our minds, static or dynamic,
qualitative or quantitative. The origin of these structures lies in the world around us, in
time, in space or in our imagination. Mathematicians are creators of forms.
In fact, mathematics seems to refuse definition. What is important is not what a
number, or a line is but their relationships, which render forms and design structures. (As
a simple example, two single points can define a line or a segment establishing various
relationships.)
The relationships, established by mathematics and space are, as will be explored
in this chapter, essential to the construction of the cities, as well as the book (defined by
Calvino “as a space”). These spaces, both the book and the cities, can be viewed in turn
119
On 1968, four years prior to publishing Città, Calvino writes to Guido Guglielmi: “Quanto mi sta a
cuore questa problematica combinatoria” (Lettere 1001).
85
from many perspectives, and imply multiple dimensions. What becomes essential is not
just what the city-spaces are made of, but how are they constructed.
The fact that mathematics is much more than “just numbers” cannot be
understated. Unfortunately, this is a common misconception. In fact within mathematics,
the branch of arithmetic is the one dealing purely with numbers. Only up to the year 500
B.C. mathematics was mainly arithmetic.
Mathematics became a full field of study in Greece. From the year 500 B.C. to
300 A.C., the Greeks became interested in the study of shapes, thus developing
Geometry. This era’s culmination has been considered to be Euclid’s Elements.
Afterwards, in the seventeenth century, Newton, and Leibniz, were responsible of
a real turning point. Individually, one in Engand, the other in Germany, correspondingly,
conceived Calculus. From then on, mathematics included the study of numbers, form,
motion, change and space (Delvin 1-2).
Later, by the end of the nineteenth century, mathematics developed further into
“pure” mathematics, by analyzing and studying the conceptual instruments used by
mathematicians. In fact mathematics does not constitute a mere tool: its principal aspects
are both intellectual and cultural (Devin 3). Mathematics has contributed to our way of
conceiving space. An example of this would be not just Euclidean but also nonEuclidean geometry discovered (or invented) in the nineteenth century; both present in
Città.
In Umberto Eco’s Il nome della rosa, the protagonist. Frate Guglielmo addresses
his disciple Adso with the following words: “Solo nelle scienze matematiche, come dice
Averrroè, si identificano le cose note per noi e quelle note in assoluto[…]La biblioteca è
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stata costruita da una mente che pensava in modo matematico, perché senza matematica
non fai labirinti” (130).
Just during this past twentieth century, the burst of mathematical activity was
remarkable. Many new areas of mathematics developed which in turn, divided
themselves into subcategories.
Mathematics is a realm of freedom where new ideas, and problems, new fields of
research continue to be discovered. Through its rigorous method, mathematics uses ideas
to create structures, to design new forms. Mathematical creativity allows freedom.
In his Lezioni americane, referring to Perec as the most inventive member of
Oulipo (“Non per niente Perec è stato il più inventivo dei partecipanti all'Oulipo fondato
dal suo maestro Raymond Queneau”). Regarding this comment Calvino adds: “Per
sfuggire all'arbitrarietà dell'esistenza, Perec come il suo protagonista ha bisogno
d'imporsi delle regole rigorose (anche se queste regole sono a loro volta arbitrarie). Ma il
miracolo è che questa poetica che si direbbe artificiosa e meccanica dà come risultato una
libertà e una ricchezza inventiva inesauribili” (“Molteplicità” 133).120
According to the German mathematician George Cantor (1845-1918), “the
essence of mathematics resides in its freedom”. Cantor was the founder of set theory, a
fundamental theory of mathematics, which establishes a one to one correspondence
between two sets, thus defining closed (finite) and open (infinite) sets. In 1901, Russell’s
Paradox showed that Cantor’s theory leads to contradiction. Two years later, in Principia
Matematica (1903), B. Russell and A. N. Whitehead tried to solve the paradox. Although
120
“In order to escape the arbitrary nature of existence, Perec, like his protagonist, is forced to impose
rigorous rules and regulations of himself, even if these rules and regulations are in turn arbitrary. But the
miracle is that the system of poetics, which might seem artificial and mechanical, produces inexhaustible
freedom and wealth of invention” (“Multiplicity” 122).
87
they did manage to avoid it, additional mathematical problems emerged. It wasn’t until
1930 when K. Gödel made the system consistent. Through mapping he created his
Incompleteness Theory. Calvino”s ars combinatoria in Città invisibili, as I intend to
examine here, integrates both mapping and Gödel’s Incompleteness Theorem.
Many branches of mathematics appear within the book integrating different
concepts used to create mental images of these cities in our mind. Some of them are:
Arithmetic, dealing with numbers; Geometry, with figures and measurements, Topology,
with shapes of surfaces; Calculus, with motion, change, space, continuity, limits, infinity;
Algebra, with relationships and functions; and finally, with reasoning, Logic, used often
for reversals, realistic paradoxes and humorous turnarounds.
Some themes emphasize particular mathematical concepts. Such is the case of the
cities and the sky in regards to mapping; recurrence and the spiral figure, in reference to
the hidden cities; or the net structure which is to be found particularly, but not
exclusively, in the thin cities. At times, some categories seem to be paired and even
reversed, intertwined- for example, the first and second themes, memory and desire:
“Desires are already memories” (8). (“I desideri sono già ricordi” 8.)
Numbers reflect the recognition of objects around us: the concepts of unity, the
concept of duality (central to this book), of triples, and so on. All of which are nothing
but abstractions. Furthermore, none of the elements denominated as geometric actually
exist: points, lines, figures, are all abstract conceptions within our minds.
Mathematics could be considered as yet another way to look at things and try to
understand them. As Calvino states in various essays such as his Lezione americane,
88
Galileo considered mathematics the language of the book of nature. The original passage
is found In Il Saggiatore, published in Rome in 1623:
La filosofia è scritta in questo grandissimo libro che continuamente ci sta
aperto innanzi agli occhi (io dico l'universo), ma non si può intendere se
prima non s'impara a intender la lingua, e conoscer i caratteri, ne’ quali è
scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli,
cerchi, ed altre figure geometriche, senza i quali mezzi è impossibile a
intenderne umanamente parola; senza questi è un aggirarsi vanamente per
un oscuro laberinto. 121
In Città, numbers and lists appear all over these fantastic cities. As mentioned, in
his book’s presentation, Calvino refers to the urban accumulation of things: “le città sono
un insieme di tante cose”. The stories display this fact.
From the very opening, the city of Diomira is one of the many whose description
begins by listing and counting multiple elements. Also, the theme of duplicity is already
present, introduced in Dorotea: “Della città di Dorotea si può parlare in due maniere: dire
che quattro torri d’alluminio s’elevano dalle sue mura fiancheggiando sette porte dal
ponte levatoio a molla che scavalca il fossato la cui acqua alimenta quattro verdi canali
che attraversano la città e la dividono in nove quartieri, ognuno di trecento case e
settecento fumaioli…” (9).122
Here, it would be appropriate to point out that words raccontare and contare are
etymologically related; they contain the same Latin root, “referring to the act of repetition
and enumeration while narrating or describing. In fact, in Spanish, the same verb, contar,
121
“It (the universe) is written in mathematical language and the characters are triangles, circles and other
geometric figures; without these it is impossible to understand a word, without these it is mere wandering
in vain around a dark maze” (The Assayer).
122
“There are two ways of describing the city of Dorothea: you can say that four aluminum towers rise
from its walls flanking seven gates with spring- operated drawbridges that span the moat whose water feeds
four green canals which cross the city, dividing it into nine quarters, each with three hundred houses and
seven hundred chimneys…”(9).
89
is used for both counting numbers and telling stories. (In English, an equivalent could be
tell and toll.) Besides, to “e-nu-me-rate”, also from Latin means to count off or name, one
by one, to list; other synonyms are to itemize, tally, recite, mention, cite. Similarly, the
noun account is defined as a report, description or event; other synonyms include
statement, calculation, reckoning and score.
This counting of things appears constantly within the cities; multiple city
descriptions begin with the notion of travel. Listing serves not just as a descriptive
technique, but follows the flow of the narrative, as it seems to emphasize the traveling
rhythm of the book.
Repetitively, the opening lines emphasize this idea, giving a particular pace to the
book. These opening lines along with the framing dialogue between Marco Polo and
Kublai Khan mark the continuity of the narrative, as well as its pauses, its intervals. Thus
perceived, they resemble bookmarks (or intervals in a musical piece) within this bookspace, as they guide the reader in his imaginary travel through these fantastic cities.
The first city begins with: “Partendo di là e andando tre giornate verso levante”
(7) (”leaving there and proceeding three days towards the East”) (7); a few pages later,
again, one encounters the city of Diomira: “Di capo a tre giornate, andando verso
mezzodì…” (12). (“At the end of three days, moving southward…”12). The feeling of
travel is portrayed by the use of direction and measurement of time. These indications
resemble those, which mathematically compose a vector: they seem to include both
direction and magnitude. A vector (Latin word meaning “carrier”) is a mathematical tool
endowed with both magnitude and direction, used not only in physics and geometry, but
also for all types of navigation, (such as sailing or flying.) However the exactitude is
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apparent; the measurements are vague, as “departing from there” refers to an unspecified
place, an indeterminate point of departure, “there” (là), which blends in with the apparent
precision of “three days towards the East”. Walking for “three days” does not give us any
exact distance, since “days” are units of time: “Partendosi di là e andando tre giornate
verso levante, l’uomo si trova a Diomira, città con sessanta cupole d’argento, statue in
bronzo di tutti gli dei, vie lastricate in stagno, un teatro di cristallo, un gallo d’oro che
canta ogni mattina su una torre” (7). (“Leaving there and proceeding for three days
toward the east, you reach Diomira, a city with sixty silver domes, bronze statues of all
the gods, streets paved with lead, a crystal theater, a golden cock that crows each morning
on a tower” 7).
Other “entrances” in the cities do not comprise “vector” quantities, since they do
not include a direction; yet, the counting of days continues to grant a traveling rhythm to
the narrative: “Di là, dopo sei giorni e sette notti” (From there after six days and seven
nights”) gets one into Zobeide (45); or, “Dopo aver marciato sette giorni attraverso”
(“After seven day’s march through woodland”) leads one straight into the central city of
the book: Bauci 77). By the end of this story of Dorotea, the double description
transforms the city into one with multiple entrances:” ma ora so che questa è solo una
delle tante vie che mi si aprivano quella mattina a Dorotea” (9).123
As previously discussed, multiplicity, the theme and title of the fifth of Calvino’s
Six Memos, becomes central in Città. From the index, the itinerary, we encounter the
elements necessary for the creation of the book space. Within the cities, duplicity appears
123
“But now I know this path is only one of the many that opened before me on that morning in
Dorothea." (9)
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to often multiply itself sustaining a network of relationships. This multiplicity of doubles,
as will be examined, leads to other mathematical concepts, such as the net.
In Melania “each time you enter the square, you find yourself caught in a
dialogue”. Even after years, if you return, you will find the same, identical dialogue going
on. However, there is an exception: it may happen, that for each person living in the city
of Melania, their part or role may be doubled, multiplied and even multiplied again (or
squared): “Capita alle volte che un solo dialogante sostenga nello stesso tempo due o più
parti: tiranno, benefattore, messaggero; o che una parte sia sdoppiata, moltiplicata,
attribuita a cento, a mille abitanti di Melania: tremila per l’ipocrita, trentamila per lo
scroccone, centomila figli di re caduti in bassa fortuna che attendono il riconoscimento”
(80-81).124
In Procopia, numbers are used to demonstrate, not without humor, the exponential
growth of the city:125
Sono sicuro che la prima volta non si vedeva nessuno; è stato solo l'anno
dopo che, a un movimento tra le foglie, ho potuto distinguere una faccia
tonda e piatta che rosicchiava una pannocchia. Dopo un anno erano in tre
sul muretto, e al mio ritorno ce ne vidi sei, seduti in fila, con le mani sui
ginocchi e qualche sorba in un piatto. Ogni anno, appena entrato nella
stanza, alzavo la tendina e contavo alcune facce in più: sedici, compresi
quelli giù nel fosso; ventinove, di cui otto appollaiati sul sorbo;
quarantasette senza contare quelli nel pollaio. Si somigliano, sembrano
gentili… (146).126
124
“At times it may happen that a sole person will simultaneously take on two or more roles-tyrant,
benefactor, messenger--or one role may be doubled, multiplied, assigned to a hundred, a thousand
inhabitants of Melania: three thousand for the hypocrite, thirty thousand for the sponger, a hundred
thousand king’s sons fallen in low estate and awaiting recognition” (80-81).
125
126
Exponential growth is further developed through the game of chess, as will be examined later.
“The first time I am sure there was no one to be seen; it was only the following year that, at a
movement among the leaves, I could discern a round, flat face, gnawing on an ear of com. A year later
there were three of them on the wall, and at my return I saw six, seated in a row, with their hands on their
knees and some medlars in a dish. Each year, as soon as I entered the room, I raised the curtain and counted
92
Besides numbers and lists, multiple geometric figures appear throughout the
cities. The basic geometric unit is the point: it is the base from which all other figures,
such as lines, planes, etc. are derived. Yet a point has no dimension.
Calvino’s story, “Tutto in un punto”, develops the idea that the whole universe
was once concentrated in a single point, before expanding into space. Ironically, Calvino
describes the difficulties of daily life in a single point (Cosmicomiche Vecchie e Nuove).
This concept is playfully developed in various cities, manipulated in multiple ways.
Calvino does not waste this opportunity; starting from a point, anything is possible. In the
city of Olinda,
Chi ci va con una lente e cerca con attenzione può trovare da qualche parte
un punto non più grade d’una capocchia di spillo che a guardarlo un po’
ingrandito ci si vede dentro i tetti le antenne i lucernari i giardini le
vasche, gli striscioni attraverso le vie, i chioschi nelle piazze, il campo per
le corse dei cavalli. Quel punto non resta lì: dopo un anno lo si trova
grande come un mezzo limone, poi come un fungo porcino, poi come un
piatto da minestra. Ed ecco che diventa una città a grandezza naturale,
racchiusa dentro la città di prima: una nuova città che si fa largo in mezzo
alla città di prima e la spinge verso il fuori (129).127
The city of Leandra is described as being “protected” by two kinds of gods;
however this dual character, this binary classification acquires a humorous turn: they are
described as infinite points: “Both are too tiny to be seen and too numerous to be
counted”(78). “Gli uni e gli altri sono così piccoli che non si vedono e così numerosi che
more faces: sixteen, including those down in the ditch; twenty-nine, of whom eight were perched in the
medlar; forty-seven, besides those in the chicken house. They look alike; they seem polite” (146).
127
“If you go out with a magnifying glass and hunt carefully, you may find somewhere a point no bigger
than the head of a pin which, if you look at it slightly enlarged, reveals within itself the roofs, the antennas,
the skylights, the gardens, the pools, the streamers across the streets, the kiosks in the squares, the horseracing track. That point does not remain there: a year later you will find it the size of half a lemon, then as
large as a mushroom, then a soup plate. And then it becomes a full-size city, enclosed within the earlier
city: a new city that forces its way ahead in the earlier city and presses it toward the outside” (129).
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non si possono contare” (78).
The point is also essential for another geometric concept, concentricity.
Concentric figures can be circles, squares, or any type of polygon: but they must share a
central point. In Città, we find that Olinda is “not the only city which grows in concentric
circles” (129). (“Olinda non è certo la sola città a crescere in cerchi concentrici” 130).
We also encounter Anastasia, explicitly “a city with concentric canals watering it” (12)
(“città bagnata dai canali concentrici”12). (Concentricity, as will be seen, evolves into
more complex mathematical mappings or figures, such as the spiral.)
Points are used to indicate a location. Geometrically, this is achieved by the use of
coordinates, which create a correspondence as in a map or between concepts (such as
time and space, etc.). In a two-dimensional surface, two coordinates are used to indicate
the horizontal and vertical components: length and width, (where as, in a three
dimensional space, three coordinates are needed to allocate length, width and height).
The city of Zora “has the quality of remaining in your memory point by point […]
Zora's secret lies in the way your gaze runs over patterns following one another as in a
musical score where not a note can be altered or displaced” (15). 128
Lines, another basic geometric concept, are composed an infinity of points.
However, only two points are necessary to create a line. This does not prevent the author
from taking full advantage of its potential for design. For instance, lines have infinite
length, but no thickness; yet they make any figure possible. In the city of Procopia, after a
128
“Zora ha la proprietà di restare nella memoria punto per punto, nella successione delle vie, e delle case
lungo le vie, e delle porte e delle finestre nelle case, pur non mostrando in esse bellezze o rarità particolari.
Il suo segreto è il modo in cui la vista scorre su figure che si succedono come in una partitura musicale
nella quale non si può cambiare o spostare una sola nota” 15).
94
list of elements from a window view, the only “piece of blue sky” which can be seen is
described as a trapezoid (a quadrilateral figure which must have two parallel lines).
Ogni anno nei miei viaggi faccio sosta a Procopia e prendo alloggio nella
stessa stanza della stessa locanda. Fin dalla prima volta mi sono
soffermato a contemplare il paesaggio che si vede spostando la tendina
della finestra: un fosso, un ponte, un muretto, un albero di sorbo, un
campo di pannocchie, un roveto con le more, un pollaio, un dosso di
collina giallo, una nuvola bianca, un pezzo di cielo azzurro a forma di
trapezio (146).129
Lines appear everywhere, in many forms, curved, as in parabolas or spirals or
straight, as in threads, establishing relationships, as in a net or a web. In his book,
Painting with Words, Writing with Pictures, Franco Ricci summarizes the importance of
the line in Calvino’s writing: “Calvino considers the line at the heart of his personal
conception of his own mind…and drawn lines appear in many of his stories. They form a
filigree design of spider webs…form an unending spiral in the short story “La spirale”;
become the shifting cityscapes of Marco Polo’s Invisible cities…” (239).
In the city of Cloe, (“La città a gli scambi 2”) as in any big metropolis, people do
not know each other. They walk along the same streets, but only their eyes cross,
momentarily: “A Cloe, grande città, le persone che passano per le vie non si conoscono.
Al vedersi immaginano mille cose l’uno dell’altro, gli incontri che potrebbero avvenire
tra loro, le conversazioni, le sorprese, le carezze, i morsi. Ma nessuno saluta nessuno, gli
129 “Each year in the course of my travels I stop at Procopia and take lodgings in the same room in the
same inn. Ever since the first time I have lingered to con template the landscape to be seen by raising the
curtain at the window: a ditch, a bridge, a little wall, a medlar, a field of corn, a bramble patch' with blackberries, a chicken yard, the yellow hump of a hill, a white cloud, a stretch of blue sky shaped like a trapeze”
(146).
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sguardi s’incrociano per un secondo e poi si sfuggono, cercano altri sguardi, non si
fermano”(51).130
Ironically, in this second “trading city” nothing is really exchanged between its
inhabitants. Their imaginations draw lines and patterns, as they glance at each other.
Potential spaces are designed by fantasy: “Qualcosa corre tra loro, uno scambiarsi di
sguardi come linee che collegano una figura all’altra e disegnano frecce, stelle, triangoli,
finché tutte le combinazioni in un attimo sono esaurite, e altri personaggi entrano in
scena” (51).131
Other than in Città, among Calvino’s multiple stories, one encounters a few
centered on a line. One example is “La mano e la linea”: “Quatro favole d’Esopo per
Valerio Adami” in ”Guardando disegni e quadri” (RR III: 414-415).132 M.C. Escher’s
lithograph Drawing Hands portrays two hands on a sheet of paper appearing to draw one
another, as if to make the other one exist. This is as explained by Douglas Hofstadter as a
“strange loop”.
130
“In Chloe, a great city, the people who move through the streets are all strangers. At each encounter,
they imagine a thousand things about one another; meetings which could take place between them, a
conversation, surprises, caresses, bites. But no one greets anyone; eyes lock for a second, then dart away,
seeking other eyes, never stopping” (51).
131
“Something runs among them, an exchange of glances like lines that connect one figure with another
and draw arrows, stars, triangles, until all combinations are used up in a moment, and other characters come
on to the scene” (51).
132
“Una linea un giorno si stancò di obbedire la mano che la faceva correre avanti e indietro sulla tela a
suo capriccio,…La linea decise di giocare d’astuzia ... E la mano s’immedesimò nel disegno d’una
mano…” (RR III: 414).
“Ma una mano disegnata non esaurirà mai le possibilità d’essere della mano che disegna. Per poco
che la mano si sposti, la prospettiva cambia…Bisognava disegnare un’altra mano, poi un’altra, un’altra
ancora. La mano non sapeva più smettere di disegnare mani, di cercare se stessa nelle infinite combinazioni
di linee che definiscono in quante combinazioni di linee: la mano quando sbuccia una sigaretta, quando
s’appoggia su un bastone, quando sfoglia un libro” (RR III: 414-415).
“La linea era sicura d’essere lei ora a condurre il gioco, perché aveva catturato la mano nel
groviglio dei suoi mobili contorni: non s’accorgeva d’essere manovrata e coartata più di prima. La mano
era sicura d’aver stabilito un rapporto con la multiforme essenza di se stessa: non s’accorgeva che ora senza
la linea non saprebbe più d’esistere“ (RR III: 415).
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In Euclidean geometry, only two points are necessary to define a line. Only two
lines are necessary to create a plane; placing single non-collinear point outside the
straight line, suffices to create another dimension, that is, to leap into a two dimensional
space: a plane or a planar figure.
Lines curved or straight can in turn be used to delineate, to visualize figures, to
establish relationships. Points constitute the “nodes” of relationships, as in a net: lines are
drawn between two points to establish these correspondences.”Anche a Raissa, città
triste, corre un filo invisibile che allaccia un essere vivente a un altro per un attimo e si
disfa, poi torna a tendersi tra punti in movimento disegnando nuove rapide figure
cosicché a ogni secondo la città infelice contiene una città felice che nemmeno sa
d’esistere”(149).133
Relationships are used as means of construction. At a certain point, the story of
Zaira turns around to signal this point. Apparently, for this city, listing does not suffice as
a descriptive strategy: “Inutilmente, magnanimo Kublai, tenterò di descriverti la città di
Zaira dagli alti bastioni. Potrei dirti di quanti gradini sono le vie fatte a scale, di che sesto
gli archi dei porticati, di quali lamine di zinco sono ricoperti i tetti; ma so già che sarebbe
come non dirti nulla” (10).134
Space and time, are measured so as to construct the city from their relationship:
However, Polo’s description continues with the listing of these measurements, which turn
133
“Also in Raissa, city of sadness, there runs an invisible thread that binds one living being to another for
a moment, then unravels, then is stretched again between moving points as it draws new and rapid patterns
so that at every second the unhappy city contains a happy city unaware of its own existence" (149).
134
“In vain, greathearted Kublai, shall I attempt to describe Zaira, city of high bastions. I could tell you
how many steps make up the streets rising like stair-ways, and the degree of the arcades' curves, and what
kind of zinc scales cover the roofs; but I already know this would be the same as telling you nothing.” (10)
97
out to point to that which is exceptional. As we read or travel inside each city that which
seems to “filter in”, almost inadvertently within the city, transforms into its essence: “the
usurper, the adulterer, a tilt, the cat’s slipping into the window, the gunboat which
suddenly appears, the rips of the fish net, and finally the storytelling of the old men at the
dock” (10).
Non di questo è fatta la città, ma di relazioni tra le misure del suo spazio e
gli avvenimenti del suo passato: la distanza dal suolo d’un lampione e i
piedi penzolanti d’un usurpatore impiccato; il filo teso dal lampione alla
ringhiera di fronte e i festoni che impavesano il percorso del corteo nuziale
della regina; l’altezza di quella ringhiera e il salto dell’adultero che la
scavalca all’alba; l’inclinazione d’una grondaia e l’incedervi d’un gatto
che si infila nella stessa finestra; la linea di tiro della nave cannoniera
apparsa all’improvviso dietro il capo e la bomba che distrugge la grondaia;
gli strappi delle reti da pesca e i tre vecchi che seduti sul molo a
rammendare le reti si raccontano per la centesima volta la storia della
cannoniera dell’usurpatore, che si dice fosse un figlio adulterino della
regina, abbandonato in fasce lì sul molo (10).135
Since the combinatorial game consists in arranging an order, where various
elements are assigned a position according to certain rules, it can be visualized as a
mapping game. It establishes a relationship between things and their location (time and/or
space). This relationship is binary, a function defined by two elements. In the city of
Tamara, for instance: “If a building has no signboard or figure, its very form and the
position it occupies in the city's order suffice to indicate its function: the palace, the
135
“The city does not consist of this, but of relationships between the measurements of its space and the
events of its past: the height of a lamppost and the distance from the ground of a hanged usurper's swaying
feet; the line strung from the lamppost to the railing opposite and the festoons that decorate the course of
the queen's nuptial procession; the height of that railing and the leap of the adulterer who climbed over it at
dawn; the tilt of a guttering and a cat's progress along it as he slips into the same window; the firing range
of a gunboat which has suddenly appeared beyond the cape and the bomb that destroys the guttering; the
rips in the fish net and the three old men seated on the dock mending nets and telling each other for the
hundredth time the story of the gun boat of the usurper, who some say was the queen's illegitimate son,
abandoned in his swaddling clothes there on the dock” (10).
98
prison, the mint, the Pythagorean136 school, the brothel” (13-14). (“Se un edificio non
porta nessuna insegna o figura, la sua stessa forma e il posto che occupa nell’ordine della
città bastano a indi- carne la funzione: la reggia, la prigione, la zecca, la scuola
pitagorica, il bordello” 13-14).
Mapping as the basis of the combinatorial game and, as stated previously, is based
on binary relationships, on establishing relationships. Duality becomes fundamental and
is central in Città as well as in Calvino’s way of thinking.137 In “Exactitude”, referring to
a passage of Città, he confides to us:
The fact is that my thinking has always found itself facing two different
paths that correspond to two different types of knowledge. One path goes
into mental spaces of bodiless rationality where one may have lines that
converge, abstract forms, vectors of form. The other path goes through a
space crammed with objects and attempts to create a verbal equivalent to
that space by filling the space with words, involving a most careful,
painstaking effort to adapt what is written to what is not written, to the
sum of what is sayable, and not sayable (74).
As previously stated in I, 25, this two “paths”, this “bifurcation”, is represented in the
book by the dialogues between the Khan and Marco Polo.
Calvino makes reference to the importance of doubles in the book’s presentation,
where he explains that cities and doubles were originally a category, a theme by which he
was trying to organize the stories; yet this classification had to be subdivided into other
categories: “Alcune potevo definirle Le città duplici, ma poi mi venne meglio distribuirle
in altri gruppi” (Città vii). Thus, it is no surprise that we find traces of duplicity, as well
as its variations and transformations, throughout the book, not just in its structure, but
136
Pythagoras is one of the two mathematicians mentioned in the Città. The other one is Thales.
137
See I: 25-26.
99
also within the text. (The rubrics of each of the cities are also binary; they are the result
of the combination of the city and one of the eleven categories used to organize the book.
Many of these eleven themes are related to the concept of duplicity in a variety of ways,
creating a network of interrelationships.)
Calvino manipulates the concept of doubles to playfully develop a multiplicity of
related concepts. That is, doubles become multiplied into what appears to be an
exponential variety of forms, the duality consenting to multiple relationships: inverses,
triples, reversals, mirrors and, finally, more complex structures such as spirals and nets.
Duplicity is used, through reversal, to create mirror images as in the city of
Valdrada. Mirrors not only replicate (duplicate) but also invert images; and this
characteristic is not wasted in the creation of this city story. The city of Valdrada, built
“on the shores inversion of a lake, with houses all verandas one above the other, and high
streets whose railed parapets look out over the water (53). Valdrada’s duplicity is
constructed by mapping and inversion. But the mirror seems magical: its reflection
allows a particular dimensional view, as the traveler can see even the houses’ interiors.
Così il viaggiatore vede arrivando due città: una diritta sopra il lago e una
riflessa capovolta. Non esiste o avviene cosa nell'una Valdrada che l'altra
Valdrada non ripeta, perché la città fu costruita in modo che ogni suo
punto fosse riflesso dal suo specchio, e la Valdrada giù nell'acqua contiene
non solo tutte le scanalature e gli sbalzi delle facciate che s'elevano sopra
il lago ma anche l'interno delle stanze con i soffitti e i pavimenti, la
prospettiva dei corridoi, gli specchi degli armadi (RR II: 399).138
138
“Thus the traveler, arriving, sees two cities: one erect above the lake, and the other reflected, up-side
down. Nothing exists or happens in the one Valdrada that the other Valdrada does not repeat, because the
city was so constructed that its every point would be reflected in its mirror, and the Valdrada down in the
water contains not only all the flutings and jottings of the facades that rise above the lake, but also the
rooms' interiors with ceilings and floors, the perspective of the halls, the mirrors of the wardrobes” (53).
100
Another main binary relationship is that established with a model, which appears
implicitly and explicitly throughout the text. A model can be defined as a construction,
which serves as a plan used for the creation of another, or from which another product is
created. It can also be a representation to show the appearance of something else or for
something that allows for the investigation of the properties of a system, and in some
cases, the prediction of future outcomes. Models are also often used in analysis. In
mathematics, they theoretically reflect some aspect of a particular process or
phenomenon and enables predictions to be made about its behavior.
As means to try to depict for Kublai Khan the intricacies of his Empire, Marco
Polo explains to him that all the cities are based on a first model, which is implicit in all
the cities. His model is the city of Venice. In the opening frame of the sixth chapter,
Marco declares this model city: “Per distinguere le qualità delle altre devo partire da una
prima città. Per me è Venezia” (RR II: 412).139 All the cities described, Polo insists, are
related to Venice: “Ogni volta che descrivo una città dico qualcosa di Venezia” (RR II:
432).140 Implicitly or explicitly, Venice is present in all of the fifty-five cities.
In the city of Fedora, a museum holds the models of all the imaginary cities
conceived by Fedora’s inhabitants: “These are the forms the city could have taken if, for
one reason or another, it had not become what we see today” (32). (“Sono le forme che la
città avrebbe potuto prendere se non fosse per una ragione o per altra” 31). The museum
stands at the center of this “gray metropolis”:” … a metal building with a crystal globe in
every room. Looking at each globe, you see a blue city, the model of a different Fedora”
139
“To distinguish the other cities’ qualities, I must speak of a first city that remains implicit. For me it is
Venice” (86).
140
“Every time that I describe a city I am saying something about Venice” (86).
101
(32). (“… un palazzo di cristallo con una sfera di vetro in ogni stanza. Guardando dentro
ogni sfera si vede una città azzurra che è il modello d’un altra Fedora” 31).
The map of the Khan’s empire, says Polo, should have space to accommodate all
Fedoras, the big city and the smaller ones in the glass globes: “Not because they are
equally real, but because they are all assumptions” (31). (“Non perché tutte ugualmente
reali, ma perché tutte solo presunte” 31).
As exposed in part, the theme of maps appears and evolves throughout the book.
In the frame, we encountered a particular emphasis towards the end of the book, on
Kublai Khan’s atlas of his Empire. Within the descriptions of the cities, some of the most
explicit are found in those related to the theme of the sky (“La città e il cielo”). Within
the cities, those related to the theme of the sky, generally display mapping most
explicitly. Some of these cities seem to be constructed by a direct, one to one (point by
point) correspondence with the sky. In other words, these cities are mapped in reference
to the sky. Previously, we said that mapping consists in establishing a mathematical
function, which defines a relationship of correspondence between two spaces. Through
mapping the elements of one space are allocated in reference to a second space. The first
space, the domain, could be the model to the second, the range. This is the case in the
mapping between some cities and the sky.
Calvino points out how at the beginning of the Metamorphoses, to introduce us
the world of the celestial gods, Ovid starts by making it so similar as to be identical to
everyday Rome…Terrestrial forms and stories echo celestial forms and stories, but each
entwines the other by turns in a double spiral 141 (“Ovid and Universal Contiguity”. Why
141
“Avvicinamento non vuol dir riduzione o ironia: siamo in un universo in cui le forme riempiono
fittamente lo spazio scambiandosi continuamente qualità è dimensioni, e il fluire del tempo è riempito da
102
read the classics? 147). Regarding this theme of the mapping between the earth and the
sky, Calvino also comments on an essay by Francis Wahl: “... come la rappresentazione
del globo terracqueo comincia soltanto quando le coordinate usate per rappresentare il
cielo vengono riferite dalla Terra. I parametri celesti (asse polare e piano equatoriale,
meridiani e paralleli) trovano il loro punto d’incontro nella sfera terrestre... Abbiamo
potuto descrivere la terra solo perché vi abbiamo proiettato il cielo” (Sabbia 23).
Due to mapping duplicity is created; at times, these maps can be viewed as
replications, reproductions, reflections; furthermore, this mirroring between the heaven
and the cities reveal opposition, counter movements, ironic paradoxes which are part of
the duplicity of the whole book.
The five cities under the category of “Le città e il cielo”, (“Cities and the Sky”)
are, consecutively: Eudossia, Bersabea, Tecla, Perinzia and Andria.
In Eudossia, “which spreads both upwards and down”, there is a carpet with
symmetrical figures which maps a space in its three dimensions, “the city's true form”
(“si conserva un tappeto in cui puoi contemplare la vera forma della città” 96). Initially,
the carpet with its symmetrical and geometrical patterns seems to show nothing in
common with the city: “A prima vista nulla sembra assomigliare meno a Eudossia che il
disegno del tappeto, ordinato in figure simmetriche che ripetono i loro motivi lungo linee
rette e circolari, intessuto di gugliate dai colori splendenti, l'alternarsi delle cui trame puoi
seguire lungo tutto l'ordito” (96).142
un proliferare di racconti. Le forme e la storie terrestri ripetono forme e storie celesti ma le une e le altre
s’avvolgono a vicenda in una doppia spirale” (“ Ovidio e la continuità universale” , Sagg I: 904).
142
.
“At first sight nothing seems to resemble Eudoxia less than the design of that carpet, laid out in
symmetrical motives whose patterns are repeated along straight and circular line, interwoven with
brilliantly colored spires in a repetition that can be followed throughout the whole woof.” (96)
103
However, upon closer examination, the carpet proves to be a genuine map of the
city through correspondences and relationships, which may not be easily perceived at a
glance. “Ma se ti fermi a osservarlo con attenzione, ti persuadi che a ogni luogo l tappeto
corrisponde un luogo della città e che tutte le cose contenute nella città sono comprese
nel disegno, disposte secondo i loro veri rapporti, quali sfuggono al tuo occhio distratto
dall’andiri-vieni dal brulichio dal pigia–pigia” (96).143
Within this mapping, the importance of the point, in this case a point of
advantage, is emphasized: a point which offers a perspective from which every detail,
measurement, design can be observed, even if only implicit: “il tappeto prova che c'è un
punto dal quale la città mostra le sue vere proporzioni, lo schema geometrico implicito in
ogni suo minimo dettaglio” (“ the carpet proves that there is a point from which the city
shows its true proportions, the geometrical scheme implicit in its every, tiniest detail”).
The carpet as a map and point of reference is illustrated in detail as it serves as a guide to
find any point within the city, even within Eudossia's confusion, which becomes “evident
in the incomplete perspective you grasp”:
It is easy to get lost in Eudoxia; but when you concentrate and stare at the
carpet, you recognize the street you were seeking in crimson or indigo or
magenta thread which, on a wide loop brings you to a purple enclosure
that is your real destination. Every inhabitant of Eudoxia compares the
carpet's immobile order with his own image of the city, an anguish of his
own, and each can find, concealed among the arabesques, an answer, the
story of his life, the twists of fate (96).144
143
“But if you pause and examine it carefully, you become convinced that each place in the carpet
corresponds to a place in the city and all the things contained in the city are included in the design, arranged
according to their true relationship, which escapes your eye distracted by the bustle, the throngs, the
shoving” (96).
144
The clear allusion to Henry James’ well-known story “the Figure in the Carpet” (1896) must be pointed
out.
104
However, a question remains: since the sky and the carpet are so different, does the city
or the carpet reflect the design given by the gods? “An oracle was questioned about the
mysterious bond between two objects so dissimilar as the carpet and the sky. One of the
two objects - the oracle replied – has the form the gods gave the starry sky and the orbit
in which the worlds revolve; the other is an approximate reflection, like every human
creation” (97).
Whether the city was designed by the gods, as some think, and that the city
reflects the sky, as you may think, “the opposite conclusion is also possible: that the city
is a map which reflects the universe” (97), reversing the spaces as in a mirror image. “Ma
allo stesso modo tu puoi trarne la conclusione opposta: che la vera mappa dell'universo
sia la città d 'Eudossia così com'è, una macchia che dilaga senza forma, con vie tutte a
zig-zag, case che franano una sull'altra nel polverone, incendi, urla nel buio.” (97)145
In his essay, “Le sculture e i nomadi”, Calvino describes how upon watching
people lined up, running way from death, he is unable to find an empty space to place
himself in line: “non trovo il varco in cui potrei introdurmi per accodarmi alla fila.” At
that moment, his only consolation is to think of the carpets, since within them the
nomad’s knowledge is weaved: “Solo un pensiero mi fa sentire a mio agio: i tappeti. È
nella tessitura dei tappeti che i nomadi depositano la loro sapienza: oggetti variegati e
leggeri che si stendono sul nudo suolo dovunque ci si ferma a passare la notte e si
arrotolano al mattino per portarli via con sé insieme a tutti i propri averi sulla gobba dei
cammelli” (“Le sculture e i nomadi” 233).
145
“But you could also come to the opposite conclusion: that the true map of the universe is the city of
Eudoxia, just as it is, a stain that spreads out shapelessly, with crooked streets, houses that crumble one
upon the other amid clouds of dust, fires, screams and darkness” (97).
105
In Bersabea, the inhabitants believe that if the other, the heavenly Bersabea
becomes their model, and they will both be the same and only on: “Si tramanda a
Bersabea questa credenza: che sospesa in cielo esista un'altra Bersabea, dove si librano le
virtù e i sentimenti più elevati della città, e che se la Bersabea terrena prenderà a modello
quella celeste diventerà una cosa sola con essa” (111).146
Tecla, a city, where not much can be seen, since it appears to be under a
constantly chaotic construction, is instead well planned: the starry sky is its blue print.
- Qual è il fine d'una città in costruzione se non una città? Dov'è il
piano che seguite, il progetto?
- Te lo mostreremo appena termina la giornata; ora non possiamo
interrompere, - rispondono.
Il lavoro cessa al tramonto. Scende la notte sul cantiere. È una notte
stellata. - Ecco il progetto, - dicono. (128)147
In order to create a harmonious relationship with the sky, Perinzia was
constructed according to meticulous calculations made by astronomers:
Chiamati a dettare le norme per la fondazione di Perinzia gli astronomi
stabilirono il luogo e il giorno secondo la posizione delle stelle,
tracciarono le linee incrociate del decumano e del cardo orientate l'una
come il corso del sole e l'altra come l'asse attorno a cui ruotano i cieli,
divisero la mappa secondo le dodici case dello zodiaco in modo che ogni
tempio e ogni quartiere ricevesse il giusto influsso dalle costellazioni
146
“This belief is handed down in Beersheba: that, suspended in the heavens, there exists another
Beersheba, where the city's most elevated virtues and sentiments are poised, and that if the terrestrial
Beersheba will take the celestial one as its model the two cities will become one” (111).
147
- What meaning does your construction have?" he asks. "What is the aim of a city under construction
unless it is a city? Where is the plan you are following, the blueprint?"
- "We will show it to you as soon as the working day is over; we cannot interrupt our work now,"
they answer.
- Work stops at sunset. Darkness falls over the building site. The sky is filled with stars. "There is
the blueprint," they say (127).
106
opportune, fissarono il punto delle mura in cui aprire le porte, prevedendo
che ognuna inquadrasse un'eclisse di luna nei prossimi mille anni. Perinzia
- assicurarono - avrebbe rispecchiato l'armonia del firmamento; la ragione
della natura e la grazia degli dei avrebbero dato la fortuna ai destini degli
abitanti (144).148
Regardless of the astronomical precision with which the city was built, as it became
populated the unexpected became inevitable. “Seguendo con esattezza i calcoli degli
astronomi, Perinzia fu edificata; genti diverse vennero a popolarla”. Monsters and freaks
appeared on the streets. “Nelle vie e piazze di Perinzia oggi incontri storpi, nani, gobbi,
obesi, donne con la barba. Ma il peggio non si vede; urla gutturali si levano dalle cantine
e dai granai, dove le famiglie nascondono i figli con tre teste o sei gambe” (144).149 ”Did
the calculations lead to monsters? Were the calculations wrong? Or was this the order of
the Gods? Now the question remains: does the city reflect the sky or does the sky reflect
the city? “Gli astronomi di Perinzia si trovano di fronte a una difficile scelta: o ammettere
che tutti i loro calcoli sono sbagliati e le loro cifre non riescono la descrivere il cielo, o
148
“Summoned to lay down the rules for the foundation of Perinthia, the astronomers established the place
and the day according to the position of the stars; they drew the intersecting lines of the decumanus and the
cardo, the first oriented to the passage of the sun and the other like the axis on which the heavens turn.
They divided the map according to the twelve houses of the zodiac so that each temple and each
neighborhood would receive the proper influence of the favoring constellations; they fixed the point in the
walls where gates should be cut, foreseeing how each would frame an eclipse of the moon in the next
thousand years. Perinthia- they guaranteed- would reflect the harmony of the firmament; nature's reason
and the gods' benevolence would shape the inhabitants' destinies” (144).
149
“In Perinthia's streets and square today you encounter cripples, dwarfs, hunchbacks, obese men,
bearded women. But the worse cannot be seen; guttural howls are heard from cellars and lofts, where
families hide children with three heads or with six legs” (144). This description allows for further,
comparative study with Calvino’s “La giornata d’uno scrutatore”, (The day of a scrutineer, 1963), which
describes Turin’s Cottolengo, a hospital, originally a religious institution for those handicapped gravely
disfigured and disabled (144).
107
rivelare che l'ordine degli dei è proprio quello che si rispecchia nella città dei mostri”
(145).150
Mathematical calculations certainly seemed to have led to what escapes the system’s
rules. (This singularity echoes the city of Eudossia, where it was not clear if the map of
the universe was contained in the carpet or the city.)
Andria was also constructed by carefully following the sky as a map, to the point
of including the motion of heavenly bodies: “Con tale arte fu costruita Andria, che ogni
sua via corre seguendo l'orbita d'un pianeta e gli edifici e i luoghi della vita in comune
ripetono l'ordine delle costellazioni e la posizione degli astri più luminosi” (150).151 In
contrast, Andria’s construction shows such a flexible relationship with the sky, that any
change in the city is in agreement with the heavens and vice-versa. “Così perfetta è la
corrispondenza tra la nostra città e il cielo, - risposero, - che ogni cambiamento d'Andria
comporta qualche novità tra le stelle-.” The mapping readjusts constantly to change: “Gli
astronomi scrutano coi telescopi dopo ogni mutamento che ha luogo in Andria, e
segnalano l'esplosione d'una nova, o il passare dall'arancione al giallo d'un remoto punto
del firmamento, l'espandersi di una nebula, il curvarsi d'una spira della via lattea”. (150)
Even the calendars are “so regulated that jobs and offices and ceremonies are
arranged in a map corresponding to the firmament on that date: and thus the days on earth
150
“Perinthia’s astronomers are faced with a difficult choice. Either they must admit that all their
calculations were wrong and their figures are unable to describe the heavens, or else they must reveal that
the order of the gods is reflected exactly in the city of monsters” (145).
151
“Andria was built so artfully that its every street follows a planet's orbit, and the buildings and the
places of community life repeat the order of the constellations and the position of the most luminous stars”
(150).
108
and the nights in the sky reflect each other” (150).152
As Marco relates, it seemed that by being part of the “spiritual ease” of an
apparent “unchanging heaven”, “a meticulous clock work”, the inhabitants would take
care not to make the slightest change”. Instead, they “led me to visit a suspended street
recently opened over a bamboo grove, a shadow-theater under construction in the place
of the municipal kennels, now moved to the pavilions of the former lazaretto, abolished
when the last plague victims were cured, and, just inaugurated, a river port, a statue of
Thales, a toboggan slide (150-151). “E mi condussero a visitare una via pensile aperta di
recente sopra un bosco di bambù, un teatro delle ombre in costruzione al posto del canile
municipale, ora traslocato nei padiglioni dell'antico lazzaretto, abolito per la guarigione
degli ultimi appestati, e - appena inaugurati - un porto fluviale, una statua di Talete, una
toboga” (150-151).153
Their correspondence, relationship is such that any alteration creates a sequence
of changes, flowing in accordance to each other, creating interplay. But for this to be
achieved, constant, accurate calculations become necessary. In fact, the “virtuosity” of
Andria’s inhabitants consists on this conviction and determination - “to calculate all
possibilities”: “Convinti che ogni innovazione nella città influisca sul disegno del cielo,
prima d'ogni decisione calcolano i rischi e i vantaggi per loro e per l'insieme della città e
152
“Il calendario della città è regolato in modo che lavori e uffici e cerimonie si dispongono in una mappa
che corrisponde al firmamento in quella data: così i giorni in terra e le notti in cielo si rispecchiano” (150).
153
Thales of Miletus, a Greek philosopher and mathematician (c. 630 A.C). An innovator in his use of
mapping geometry is believed to have taught Anaximander who, in turn, became Pythagoras’ teacher.
According to Bertrand Russell, “Western philosophy began with Thales”. Thales and the aforementioned
Pythagoras are the two mathematicians mentioned in the book.
109
dei mondi” (151).154
This fifth, and last city in the category of “La città e il cielo”, seems a construct of
a perfect mapping, as it transforms itself, adjusting to any change. This dynamic
correspondence, in particular, not only implies but also exposes the fact that the changes
in the city could affect those in the universe.155
2. 2. 2 Doubling doubles
As examined, the cities within the category of “The cities and the sky” (“Le città e
il cielo”) are based on mappings, which at times can be viewed as replications,
reproductions, reflections; furthermore, this mirroring between the heaven and cities
reveal opposition, counter movements, ironic paradoxes which are part of the duplicity of
the whole book.
Other than these mappings between the cities and the sky, some others among the
fifty-five cities, stand out as explicit examples of duplicity.
The city of Zemrude presents itself as double-sided: one lower which can be
watched from above, one, from below; yet “you cannot say that one aspect of the city is
truer than the other” since “it is the mood of the beholder which gives the city of
Zemrude its form” (69): (“E ̀ l’umore di chi la guarda che dà alla città di Zemrude la sua
forma” 69). Both sides have equal “value”, leading to “balance” as in algebraic equation.
This seems to be the essence or singularity of the city.
154
“Convinced that every innovation in the city influences the sky's pattern, before taking any decision
they calculate the risks and advantages for themselves and for the city and for all worlds” (151).
155
Potentially, this could be interpreted as suggesting an ecological perspective. This could be a subject for
another study.
110
However, the emphasis is placed on perspective, as it is the way in which one
views the city that grants the city “its form”. The city can be looked from below as one
looks up or from above as one looks down. (“If you go by whistling, your nose a-tilt
behind the whistle, you will know it from below: windowsills, flapping curtains,
fountains. If you walk along hanging your head, your nails dug into the palms of your
hands, your gaze will be held on the ground, in the gutters, the manhole covers, the fish
scales, wastepaper” 69).
The concept of perspective is also developed by the city of Despina through its
double character. Despina appears as two different cities, depending on whether
approached from sea or from land: “In due modi si raggiunge Despina: per nave o per
cammello. La città si presenta differente a chi viene da terra e a chi dal mare“ (RR II:
370).156
Thus, through opposition, Despina displays the concept of limits, of boundaries
between spaces: “Ogni città riceve la sua forma dal deserto a cui si oppone; e così il
cammelliere e il marinaio vedono Despina, città di confine tra due deserti”.157 Contrast,
distinction delineates the shape of each city. The continuity of the sea is in contrast to the
discrete character of the sand.
The inverse of a double, that is a half, is also used in the construction of a city.
Sofronia is composed of “two half-cities”. However, regardless of expectation, the half
that projects motion (spinning wheels, carousel) remains permanent, whereas the other
156
“Despina can be reached in two ways: by ship or by camel. The city displays one face to the traveler
arriving overland and a different one to him who arrives by sea“ (17).
157
“Each city receives its form from the desert it opposes; and so the camel driver and the sailor see
Despina, a border city between two deserts” (18).
111
half that of stone, marble and cement is temporary. They seem to live in symbiosis, while
their roles and functions are inverted. For the city to survive in this coexistence, it must
continue to dismantle and transport itself, searching for those half empty spaces, in which
to rebuild its other half, which completes the city’s form. (Thus motion through space,
defined by change in location, requires space that allows a mobile mapping.)
Sofronia si compone di due mezze città. In una c’è il grande ottovolante
dalle ripide gobbe, la giostra con la raggiera di catene, la ruota delle
gabbie girevoli, il pozzo della morte con i motociclisti a testa in giù, la
cupola del circo col grappolo dei trapezi che pende in mezzo. L’altra
mezza città è di pietra e marmo e cemento, con la banca, gli opifici, i
palazzi, il mattatoio, la scuola e tutto il resto. Una delle mezze città è fissa,
l’altra è provvisoria e quando il tempo della sua sosta è finito la schiodano,
la smontano e la portano via, per trapiantarla nei terreni vaghi158 d’un’altra
mezza città (RR II: 409).159
Moriana, instead, is a bi-dimensional city, with an “obverse”: “you have only to
walk in a semicircle and you will come into view of Moriana’s hidden face” an expanse
of rusting sheet metal, sackcloth, planks bristling with spikes, pipes black with soot, piles
of tins, blind walls with fading signs, frames of staved- in straw chairs, ropes good only
for hanging oneself from a rotten beam (105). “Se non è al suo primo viaggio l’uomo sa
158
In “Exactitude”, Calvino comments on the word “vague”: “I might mention in passing that as far as I
know Italian is the only language in which the word vago (vague) also means “lovely”, “attractive”.
Starting out from the original meaning of “wandering”, the word vago still carries an idea of movement and
mutability, which in Italian is associated both with uncertainty and indefiniteness and with gracefulness and
pleasure“ (57). “Resta da vedere se con argomenti altrettanto convincenti non si possa difendere anche la
tesi contraria. Per esempio, Giacomo Leopardi sosteneva che il linguaggio è tanto più poetico quanto più è
vago, impreciso. (Noterò per inciso che l'italiano è l'unica lingua - credo - in cui vago significa anche
grazioso, attraente: partendo dal significato originale (wandering) la parola vago porta con sé un'idea di
movimento e mutevolezza, che(67 s'associa in italiano tanto all'incerto e all'indefinito quanto alla grazia,
alla piacevolezza)” (Lezioni 67-68).
159
“The city of Sophronia is made up of two half-cities. In one there is the great roller coaster with its steep
humps, the carousel with its chain spokes, the Ferris wheel of spinning cages, the death-ride with crouching
motorcyclists, the big top with the clump of trapezes hanging in the middle. The other half-city is of stone
and marble and cement, with the bank, the factories, the palaces, the slaughterhouse, the school, and all the
rest. One of the half-cities is permanent, the other is temporary, and when the period of its sojourn is over,
they uproot it, dismantle it, and take it off, transplanting it to the vacant lots of another half-city” (63).
112
già che le città come questa hanno un rovescio: basta percorrere un semicerchio e si avrà
in vista la faccia nascosta di Moriana, una distesa di lamiera arrugginita, tela di sacco,
assi irte di chiodi, tubi neri di fuliggine, mucchi di barattoli, muri ciechi con scritte stinte,
telai di sedie spagliate, corde buone solo per impiccarsi a un trave marcio” (RR II: 441).
Moriana is a flat, planar city.160 By definition, a plane has only length and width, but no
depth. But, curiously, this city with only two dimensions uses perspective to multiply
itself. Analogously in mathematics, only one point outside a plane, that is, a non-coplanar
point (like a point of perspective) is necessary to create a three dimensional space or
figure.“Da una parte all’altra la città sembra continui in prospettiva moltiplicando il suo
repertorio d’immagini: invece non ha spessore, consiste solo in un diritto e in un
rovescio, come un foglio di carta, con una figura di qua e una di là, che non possono
staccarsi né guardarsi” (105).161
Besides, this city, compared to a piece of paper, could also be seen as a page,
within this book-city, having figures or images on each of its opposite sides, which do not
face one another; yet, they can each be multiplied by the reader’s interpretations or views,
that is, through different perspectives.
This flat, completely horizontal city finds its inverse in Argia, which displays
another kind of duplicity, constructed through inversion. Argia is buried in depth: “Ciò
che fa Argia diversa dalle altre città è che invece d’aria ha terra” (126). (What makes
Argia different from other cities is that it has earth instead of air” 126). The concept of
160
161
Moriana recalls Edwin Abbot’s Flatland (1984).
“From one part to the other, the city seems to continue, in perspective, multiplying its repertory of
images: but instead it has no thickness, it consists only of a face and an obverse, like a sheet of paper, with
a figure on either side, which can neither be separated or look at each other” (105).
113
depth opposes Moriana’s flatness: “Le vie sono completamente interrate, le stanze sono
piene d’argilla fino al soffitto, sulle scale si posa un’altra scala in negativo, sopra i tetti
delle case gravano strati di terreno roccioso come cieli con le nuvole” (127).162
Argia, is explicitly invisible, since it is a city that can only be “heard” (or read,
like the others). The one thing, which can be perceived, is a casual slamming of a
door,”at night, putting your ear to the ground, you can sometimes hear a door slam”
(126), escaping from within the darkness of the city into another “darkness”, that of the
night. “From up here, nothing of Argia can be seen; some say, ‘It's down below there,’
and we can only believe them. The place is deserted” (126).
Not only is the air replaced by earth. Even the stairway is reversed in this buried
city: its negative placed over itself. To counteract the gloomy aspect of this tomb, Polo’s
description turns humorously into speculations on the unlikely possibility of motion
allowed to or remaining for the city’s inhabitants. “Se gli abitanti possono girare per la
città allargando i cunicoli dei vermi e le fessure in cui s’insinuano le radici non lo
sappiamo: l’umidità sfascia i corpi e lascia loro poche forze; conviene che restino fermi e
distesi, tanto è buio” (127).163
The city of Maurilia, the fifth and last of the series “La città e la memoria”
(“Cities and Memory”) manages to double itself constantly and simultaneously, existing
162
“The streets are completely filled with dirt, clay packs the rooms to the ceiling, on every stair another
stairway is set in negative, over the roofs of the houses hang layers of rocky terrain like skies with clouds”
(126).
163
“We do not know if the inhabitants can move about the city, widening the worm tunnels and the
crevices where roots twist: the dampness destroys people's bodies and they have scant strength; everyone is
better off remaining still, prone; anyway, it is dark” (126). The translation, by William Weaver, could
emphasize the intensity of darkness: “tanto è buio”. Perhaps a more accurate meaning would be “it is so
dark” or “such is the darkness”.
114
as the present city as well as its past one, through its own postcards:”A Maurilia, il
viaggiatore è invitato a visitare la città e nello stesso tempo a osservare certe vecchie
cartoline illustrate che la rappresentano com’era prima” (30).164
As yet another city based on duplicity, Maurilia’s very existence relies in coexistence. The city and the postcards happen to be “there”, at the same time, (trying to
share a space). But there is a lack of correspondence: the only thing that they have in
common, that which persists is a name (without it this invisible city could not occupy
even an imaginary space). This name, Maurilia, can potentially correspond to different
cities: its mapping becomes more complex: “It is pointless to ask whether the new ones
are better or worse than the old, since there is no connection between them, just as the old
post cards do not depict Maurilia as it was, but a different city which, by chance, was
called Maurilia, like this one” (31).
Simultaneity, or agreement in time, and sometimes space, complicates mapping.
The city of Marozia consists in two cities (“quella del topo e quella delle rondine”) at the
same time (RR II: 489):”Una Sibilla, interrogata sul destino di Marozia, disse: – Vedo
due città: una del topo, una della rondine” 154. (“A Sibyl, questioned about Marozia's
fate, said: I see two cities: one of the rats, one of the swallow’”154).
Laudomia (Cities and the dead 5) could have just been another double city: one
for life and another for death. (In this case, it is the names of the inhabitants, which are
shared, leading to continuity). However, the singularity of this city consists in having a
triple: the city of those yet to be born: Ogni città, come Laudomia, ha al suo fianco
164
“In Maurilia, the traveler is invited to visit the city and, at the same time, to examine some old post cards
that show it as it used to be” (30).
115
un’altra città i cui abitanti si chiamano con gli stessi nomi: è la Laudomia dei morti, il
cimitero. Ma la speciale dote di Laudomia è d’essere, oltre che doppia, tripla, cioè di
comprendere una terza Laudomia che è quella dei non nati (140).165
However, the presumption that the number of inhabitants in this third replica may
be infinite gives way to a disproportionate allocation of space: its dimensions are
limitless, in every sense and direction: they can tend to a maximum or a minimum size, as
anything can be imagined. (Incidentally, in Calculus, when a “function” (such as
mapping) shows these characteristics, they are considered “undefined”.)
Giustamente Laudomia assegna una residenza altrettanto vasta a coloro che
ancora devono nascere; certo lo spazio non è in proporzione al loro numero che si
suppone sterminato, ma essendo un luogo vuoto, circondato da un’architettura
tutta nicchie e rientranze e scanalature, e potendosi attribuire ai non nati la
dimensione che si vuole, pensarli grandi come topi o come bachi da seta o come
formiche o uova di formica, nulla vieta d’immaginarli ritti o accoccolati su ogni
aggetto (142).166
Indeed, the trace of irony is not casual. In this description, which approximates
the mistakes of urban planning in modern cities; humor turns the story around as the
duplication of the city, its copying itself, creates the problematic of overlapping,
occupying the same space, one on top of another (overpopulation). “Le proprietà della
città doppia sono note. Più la Laudomia dei vivi s’affolla e si dilata, più cresce la distesa
delle tombe fuori dalle mura”.167 “Le vie della Laudomia dei morti sono larghe appena
165
“Like Laudomia, every city has at its side another city whose inhabitants are called by the same names:
it is the Laudomia of the dead, the cemetery. But Laudomia's special faculty is that of being not only
double, but also triple; it comprehends, in short, a third Laudomia, the city of the unborn” (140).
166
“Rightly, Laudomia assigns an equally vast residence to those who are still to be born. Naturally the
space is not in proportion to their number, which is presumably infinite, but since the area is empty…”
(141).
167 “The
properties of the double city are well known. The more the Laudomia of the living becomes
crowded and expanded, the more the expanse of tombs increases beyond the walls” (141).
116
quanto basta perché vi giri il carro del becchino, e vi s’affacciano edifici senza finestre;
ma il tracciato delle vie e l’ordine delle dimore ripete quello della Laudomia viva, e come
in essa le famiglie stanno sempre più pigiate, in fitti loculi sovrapposti.”168
As the inhabitants living Laudomia search, or literally dig, for answers in their
“dead” copy, they only find potential pasts, fragments, contradictions, and
disillusionment.“E per sentirsi sicura la Laudomia viva ha bisogno di cercare nella
Laudomia dei morti la spiegazione di se stessa, anche a rischio di trovarvi di più o di
meno: spiegazioni per più d’una Laudomia, per città diverse che potevano essere e non
sono state, o ragioni parziali, contraddittorie, delusive” (141).169
The ironic tone increases its pitch, as the “living” continue their self-exploration.
Travel to their third copy, that of the “unborn”, is performed in secrecy and silence, as if
it were a prohibited or sacred “empty” space: yet their questioning reveals their true
interest regards only “themselves”, that is what they perceive as a problem and a reality,
without any concern for future planning; ironically, the “pattern” continues to repeat
itself, ad infinitum. “I viventi di Laudomia frequentano la casa dei non nati
interrogandoli; i passi risuonano sotto le volte vuote; le domande si formulano in silenzio:
168
“The streets of the Laudomia of the dead are just wide enough to allow the gravedigger's cart to pass,
and many windowless buildings look out on them; but the pattern of the streets and the arrangement of the
dwellings repeat those of the living Laudomia, and in both, families are more and more crowded together,
in compartments crammed one above the other” (141).
169
“And to feel sure of itself, the living Laudomia has to seek in the Laudomia of the dead the explanation
of itself, even at the risk of finding more there, or less: explanations for more than one Laudomia, for
different cities that could have been and were not, or reasons that are incomplete, contradictory,
disappointing” (141).
117
ed è sempre di sé che chiedono i vivi, e non di quelli che verranno”.170
The possibility of the city’s disappearance makes its entrance in Polo’s
description, again through an analogy based on a double shape or figure: the hourglass.
Here, the discrete character of the sand becomes part of the continuous flow of time, from
life to death. The “other” bulb of the hourglass that of those to be born, also, the third
city’s number of inhabitants, however, sets the limit. “E allora la Laudomia dei morti e
quella dei non nati sono come le due ampolle d’una clessidra che non si rovescia, ogni
passaggio tra la nascita e la morte è un granello di sabbia che attraversa la strozzatura, e
ci sarà un ultimo abitante di Laudomia a nascere, un ultimo granello a cadere che ora è
qui che aspetta in cima al mucchio” (143).171
Eusapia’s construction includes a subterranean double of itself (RR II: 452). This
urban planning is not only well justified, – to alleviate the harshness of that jump (or
sudden transition) “from life to death”. (This is the third and thus central city within the
category “Cities and the Dead”): “Non c’è città più di Eusapia propensa a godere la vita e
a sfuggire gli affanni. E perché il salto dalla vita alla morte sia meno brusco, gli abitanti
hanno costruito una copia identica della loro città sottoterra “(109).172
Moreover, the reason for the existence and creation of this necropolis is
170
“Those living in Laudomia frequent the house of the unborn to interrogate them: footsteps echo beneath
the hollow domes; the questions are asked in silence; and it is always about themselves that the living ask,
not about those who are to come. “(141).
171
“Then the Laudomia of the dead and that of the unborn are like the two bulbs of an hourglass which is
not turned over; each passage between birth and death is a grain of sand that passes the neck, and there will
be a last inhabitant of Laudomia born, a last grain to fall, which is now at the top of the pile, waiting”
(143).
172
“No city is more inclined than Eusapia to enjoy life and flee care. And to make the leap from life to
death less abrupt, the inhabitants have constructed an identical copy of their city, underground” (109).
118
surrounded by humor as: “The job of accompanying the dead down below and arranging
them in the desired place is assigned to a confraternity of hooded brothers (109).
(“L’incombenza di accompagnare giù i morti e sistemarli al posto voluto è affidata a una
confraternita di incappucciati.”)
Once the absolute authority of Eusapia’s “brotherhood” is established in regards
to the dead city,173 the “other” city, its double, the narrative swiftly turns around,
ironically pretending to be based on hearsay or rumors rather than an account of a city
been visited (even if it is, just like all the others, invisible.) The story becomes
humorously guided by gossip: “rumor has it that some of them are already dead but
continue going up and down”. This is emphasized by the repetitive, rhythmic pattern of
initiating the descriptive paragraphs with the same word: “Dicono” (“They say”).
Dicono che la stessa confraternita esiste tra i morti e che non manca di dar
loro una mano; gli incappucciati dopo morti continueranno nello stesso
ufficio anche nell’altra Eusapia; lasciano credere che alcuni di loro siano
già morti e continuino a andare su e giù. Certo, l’autorità di questa
congregazione sull’Eusapia dei vivi è molto estesa.
Dicono che ogni volta che scendono trovano qualcosa di cambiato
nell’Eusapia di sotto; i morti apportano innovazioni alla loro città; non
molte, ma certo frutto di riflessione ponderata, non di capricci passeggeri.
Da un anno all’altro, dicono, l’Eusapia dei morti non si riconosce. E i vivi,
per non essere da meno, tutto quello che gli incappucciati raccontano delle
novità dei morti, vogliono farlo anche loro. Così l’Eusapia dei vivi ha
preso a copiare la sua copia sotterranea.
Dicono che questo non è solo adesso che accade: in realtà sarebbero stati i
morti a costruire l’Eusapia di sopra a somiglianza della loro città. Dicono
che nelle due città gemelle non ci sia più modo di sapere quali sono i vivi
173
“Nessun altro ha accesso all’Eusapia dei morti e tutto quello che si sa di laggiù si sa di loro.” (“No one
else has access to the Eusapia of the dead and everything known about it has been learned from them” 109110).
119
e quali i morti (110).174
Humor becomes intensified, through a repetition, which describes a repetition:
double city continues to be doubled by the “original”, as the city’s inhabitants continue
doubling its alternative duplicate. “And the living, to keep up with them, also want to do
everything that the hooded brothers tell them about the novelties of the dead. So the
Eusapia of the living has taken to copying its underground copy” (110).175
This continuous reversal occurs, until there is no distinction between the two
cities, they become identical: “Dicono che nelle due città gemelle non ci sia più modo di
sapere quali sono i vivi e quali i morti” (110). (“They say that in the twin cities there is no
longer any way of knowing who is alive and who is dead” (110).
As we travel through the cities, we find how doubles can, not only oppose each
other creating polarity (as in the dialogue between the Khan and Polo), but also converge
into a continuum (as in “Le città continue”).
– Mi sembra che tu riconosci meglio le città sull’atlante che a
visitarle di persona, – dice a Marco l’imperatore richiudendo il
libro di scatto.
E Polo: – Viaggiando ci s’accorge che le differenze si perdono:
ogni città va somigliando a tutte le città, i luoghi si scambiano
174
“They say that every time they go below they find something changed in the lower Eusapia; the dead
make innovations in their city; not many, but surely the fruit of sober reflection, not passing whims. From
one year to the next they say, the Eusapia of the dead becomes unrecognizable […] They say that this has
not just now begun to happen: actually it was the dead who built the upper Eusapia, in the image of their
city” (110).
175
This ironic depiction of the social and religious aspects of the city reflects Calvino’s definition of humor
in relationship to lightness, that is, “…contemplare il proprio dramma come dal di fuori e dissolverlo in
malinconia e ironia. [...]La gravità senza peso di cui ho parlato a proposito di Cavalcanti riaffiora nell'epoca
di Cervantes e di Shakespeare: è quella speciale connessione tra melanconia e umorismo,… così lo humour
è il comico che ha perso la pesantezza corporea (quella dimensione della carnalità umana che pur fa grandi
Boccaccio e Rabelais) e mette in dubbio l'io e il mondo e tutta la rete di relazioni che li costituiscono”
(“Leggerezza” 24-25).
120
forma ordine distanze, un pulviscolo informe invade i continenti. Il
tuo atlante custodisce intatte le differenze: quell’assortimento di
qualità che sono come le lettere del nom(139).176
In contrast to creating a map, traveling through the cities (reading the book) not
only creates a web of fantastic spaces in our minds but, as differences tend to disappear,
that which seemed to be discrete, discontinuous, like the cities, begins to appear as
continuous. (An example is the mentioned city of Trude, one of the five continuous cities,
where this theme is developed.) Also, the ambiguity in some mappings (cities and names,
cities and signs) create confusion, particularly because the relationships established are
not one-to-one, but one-to-many, as for example, the city of Ipazia.177
Doubles replicate themselves multiplying into a variety of relationships, which
delineate different structures, shapes, forms. As these become more complex, mapping
acquires intricacies. Such is the case of the net, which is not only the structural strategy of
the book space, but also essential to the filiformi central cities of lightness: “ragnatele di
rapporti intricate che cercano una forma”.
176
- I think you recognize cities better on the atlas than when you visit them in person, - the emperor says
to Marco, snapping the volume shut.
- And Polo answers, -Traveling, you realize that differences are lost: each city takes to resembling all cities,
places exchange their form, order, distances, a shapeless dust cloud invades the continents. Your atlas
preserves the differences intact: that assortment of qualities which are like the letters in a name (137).
177
Ipazia was a fourth century Greek mathematician, astronomer and Neoplatonic philosopher. I am
interested in continuing a study of the names of Calvino’s invisible cities, in order to find other meanings,
connotations, allusions and ethimological implications.
121
Chapter Three: Space in Search of a Form
“Le città non sono altre che la forma nel tempo” (RR II: 1365).178
Calvino’s constant search for form –an intrinsic aspect of his writing and his
combinatorial art- also comprehends a counterpart. He confides: “My discomfort that
arises from the loss of form that I notice in life…” (Six Memos 57) (“Il mio disagio è per
la perdita di forma che constato nella vita…”) (Lezioni 67) (Also present in the opening
of Città and in Il cavaliere inesistente).179
The combinatorial game presents both a mathematical challenge and a search. Its
challenge or sfida is that of finding a solution among the multiplicity of possibilities, to
find a way out of this labyrinth created by alternate paths, to counteract the weight, the
heaviness of the system. Its search consists, therefore, in the achievement of new forms,
mappings that not only attempt to attain or approximate images of lightness - moving,
ascending, suspended shapes- but also reflect their innate elusive, twisty, intricate
178
179
This quote appears in “Le ciità invisibili” by Mario Barenghi on “Notte e notizie dei testi” RR II.
“Lo scorse sotto un pino, seduto per terra, che disponeva le piccole pigne cadute al suolo secondo un
disegni regolare, un triangolo isoscele. A quell’ora dell’alba Agilulfo aveva sempre bisogno d’applicarsi a
un esercizio d’esattezza: contare oggetti, ordinarli in figure geometriche, risolvere problemi d’aritmetica. E
l’ora in cui le cose perdono la consistenza…l’ora in cui meno si è sicuri dell’esistenza del mondo…il
mondo intorno sfumava nell’incerto, nell’ambiguo…Allora si metteva a contare: foglie, pietre, lance,
pigne, qualsiasi cosa avesse davanti. O a metterle in fila, a ordinarle in quadrati o piramidi. L’applicarsi a
queste esatte occupazioni gli permetteva di vincere il malessere, d’assorbire la scontentezza, l’inquietudine
e il marasma, e di riprendere la lucidità e compostezza abituali “ (Il cavaliere inesistente 18 -19).
At the opening of Città:“Nella vita degli imperatori c’è un momento, che segue all’orgoglio per
l’ampiezza sterminata dei territori che abbiamo conquistato, alla malinconia e al sollievo di sapere che
presto rinunceremo a conoscerli e a comprenderli; un senso come di vuoto che ci prende una sera con
l’odore degli elefanti dopo la pioggia e della cenere di sandalo che si raffredda nei bracieri; una vertigine
che fa tremare i fiumi e le montagne istoriati sulla fulva groppa dei planisferi, arrotola uno sull’altro i
dispacci che ci annunciano il franare degli ultimi eserciti nemici di sconfitta in sconfitta, e scrosta la
ceralacca dei sigilli di re mai sentiti nominare che implorano la protezione delle nostre armate avanzanti in
cambio di tributi annuali in metalli preziosi, pelli conciate e gusci di testuggine: è il momento disperato in
cui si scopre che quest’impero che ci era sembrato la somma di tutte le meraviglie è uno sfacelo senza fine
né forma che la sua corruzione è troppo incancrenita perché il nostro scettro possa mettervi riparo, che il
trionfo sui sovrani avversari ci ha fatto eredi della loro lunga rovina. Solo nei resoconti di Marco Polo,
Kublai Kan riusciva a discernere, attraverso le muraglie e le torri destinate a crollare, la filigrana d’un
disegno così sottile da sfuggire al morso delle termiti" (3).
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character, capable of generating a coexistence between that which is finite, discrete and
that which is continuous, infinite.
This chapter will be focused on examining how the mathematical concepts
involved in the combinatorial game manage to design different forms – regardless of their
complexities- in the book space of Le città invisibili. These combinatorial potential forms
– mostly based on mobility and lightness – try to achieve and approach further intricacies
involving continuity and evasiveness (the spiral), multiplicity and unity, through the light,
ephemeral and the threadlike (the net). Visualizing the invisible cities comprises
recognizing and understanding these forms in potentia: to be able to imagine what is
possible, what can become of the present (and past) city.180
In this chapter, exploring and investigating the search for form in space, the
concept of mobility is fundamental. With this in mind, the game of chess will be
examined first. What begins with a rigid grid with limited elements evolves into an
exponential process of unlimited combinations
Secondly, we will analyze more in depth the mathematical relationships
established by a model, namely the more complex aspects of this essentially binary yet,
combinatorial mapping. This travel through the study of doubles leads to that of networks
and other intricacies.
180
“What Ulysses saves from the power of the lotus, from Circe’s drugs, and from the Siren’s song, is not
just the past or the future. Memory truly counts – for an individual, a society, a culture – only if it holds
together the imprint of the past and the plan for the future, if it allows one to do things without forgetting
what one wanted to do, and to become without ceasing to be, to be without ceasing to become” (Corriere
della sera, 10 August 1975). (“La memoria conta veramente – per gli individui, la collettività, le civiltàsolo se si tiene insieme l’impronta del passato e il progetto del futuro; se permette fare senza dimenticare
quel che si voleva fare, di diventare senza smettere di essere, di essere senza smettere di diventare”
Corriere della sera, 10 agosto 1975).
123
Nets create relationships, connections, entanglements between points. They trace
and allow motion in space. Spirals convey forms in motion. As they unwind, unravel,
disentangle around a point, they move away from this concentric point, evolving into an
ever changing form, unfolding creativity. Its relation to space and time, the spiral allows
its structure to expand or regress: the motion can be viewed in two directions, inward or
outward, thus representing both unity and multiplicity, respectively. Due to these
characteristics, the spiral expresses creativity.181
Finally, these complexities will be, if not completely resolved, at least
conceptualized as a whole, through various intrinsically mathematical approaches and
perspectives. Departing from the center of the book - where the theme of lightness shares
this pivotal position, becoming concentric with the images of suspended spider-web,
ascending, cities - one can, from a distant or void place (Bauci), also trace spiral shapes,
which through opening or closing lead to the appearance of evasiveness. Fleeting
elements manage to escape the system, but still wait with all their potential to be
unveiled, reinvented, rediscovered, thus triggering endless imagination.
In Six Memos, Calvino states:
Still there is another definition in which I recognize myself fully and that
is the imagination as a repertory of what is potential, what is hypothetical,
of what does not exist and has never existed, and perhaps never exist but
might have existed… The spiritus phantasticus is, according to Giordano
Bruno, mundus quidem et sinus inexplebilis formarum et specierum; that
is, a world or a gulf, never saturable, of forms and images. So, then I
181
Evangelista Torricelli (1608 -1647) succeeded Galileo at Florence as a mathematician to the grand duke
of Tuscany... He computed many areas, volumes, tangents, discussed the cycloid, performed… Among
other breakthroughs, in “De infinitis spirabilus” “discovered that the arc length of a logarithmic spiral
remains finite when it winds an infinite number of times around its asymptotic point” (1643) (A Source
Book in Mathematics 12000-1800. 1990, 227, 231).
124
believe, that to draw on this gulf of potential multiplicity is indispensable
for any kind of knowledge (91).182
3.1 Motion in Space: The Game of Chess
Just as no chess player will ever live long enough to exhaust all the
combinations of possible moves for the thirty-two pieces on the
chessboard, so we know (given the fact that our minds are chessboards
with hundreds of billions of pieces) that not even a lifetime lasting as long
as the universe would one ever manage to make all possible plays. But we
also know that all these are implicit in the overall code of mental plays,
according to the rules by which each of us, from one moment to the next,
formulates his thoughts […] (Uses of Literature 8-9)183
As examined, the index constitutes both a map and a game. Mapping appears to
be not just fundamental to the structure of the book as a whole but also to the
construction of its individual parts - the cities as well as the frame - since it is, as
explained, the essence of the combinatorial game. (In fact mapping could also be viewed
as a game itself; that is, a game that is played by assigning or adjusting, reorganizing
locations and meanings). The theme of maps, acquires particular significance, as we saw
on the previous chapters, both in the construction of cities and at the end of the book’s
frame, where the discussion revolves on the Khan’s magic atlas.
Many mathematical techniques and strategies are implicit through duplicity,
reversals, etc. Explicitly, we encounter, not just the combinatorial game, but also that of
182 “Ma c'è un'altra definizione in cui mi riconosco pienamente ed è l'immaginazione come repertorio del
potenziale, dell'ipotetico, di ciò che non è né è stato né forse sarà ma che avrebbe potuto essere...
Lo spiritus phantasticus secondo Giordano Bruno è "mundus quidem et sinus inexplebilis formarum et
specierum" (un mondo o un golfo, mai saturabile, di forme e d'immagini). Ecco, io credo che attingere a
questo golfo della molteplicitá potenziale sia indispensabili per ogni forma di conoscenza” (102).
183
“Sappiamo che, come nessun giocatore di scacchi potrà vivere abbastanza a lungo per esaurire le
combinazioni nelle possibili mosse dei trentadue pezzi sulla scacchiera, così- dato che la nostra mente è una
scacchiera in cui sono messi in gioco centinaia di miliardi di pezzi- neppure I una vita che durasse quanto
l’universo s’arriverebbe a giocarne tutte le partite possibili. Ma sappiamo anche che tutte le partite son
implicite nel codice generale delle partite mentali, attraverso il quale ognuno di noi formula di momento in
momento i suoi pensieri…” (Saggi I: 210).
125
the game of chess (emphasized on the eighth chapter.) Maps and games develop from the
beginning, throughout and at the conclusion of the book. They both allocate and embody
patterns in space. Mapping is at the core of the combinatorial game and, therefore, that of
chess.184
A link between mathematics and chess is to be found in the abstractness, the
reductionism involved in the game. Also the pursuit and evasion, the chase and the
avoidance intrinsic to the game are essentially calculations, which can be subject to
mathematical graphing and strategic planning. The game consists in a combinatorial
search of possible moves. Constant motion within two opponents results from accurate
mathematical pre-calculations, which are but attempts to grasp a winning pattern in this
ever-changing formations or arrangement in a contrived space. These arrangements of
chessmen are decisively combinatorial. The game model seeks to represent its potential.
The creativity of the game involves recognizing patterns and imagining new ones.
Chess, essentially a combinatorial game, allows mobility within its space. A few
elements, under rigid rules, not only permit a multiplicity of possible moves but,
furthermore, grant exponential growth. Simplifying infinitely large amounts is required as
well as mental agility. Abstractions, reduction, discrete elements, patterns in motion
constitute elements which link the notion of lightness to mathematical concepts such as
agility, dexterity and precision.
Furthermore, the game of chess can be considered as a series of deliberate
approximations, premeditated estimates, and calculated guesses. Moves are intended,
184
Chess has haunted the mind of mathematicians through ages, due to its combinatorial potential. Some
famous mathematicians who studied chess where Leonard Euler (1707-1783), Alan Turing (1912- 1954),
Norbert Weiner (1894-1964), Luca Pacioli (1445-1517), Lewis Carroll and Francois Le Lyonnais (19011984), cofounder of Oulipo with Raymond Queneau.
126
designed, planned, analyzed and computed, over and over in a zigzag motion analogous
to that of Città. “Here too, the example of science can be of use…in considering each and
every result as being part of a possibly infinite series of approximations.”(“Two
Interviews on Science and Literature”. Uses of Literature 38). In “Philosophy and
Literature”, Calvino imparts the game of chess as a tool for philosophers to “reduce the
variety of existing things to a spider-web of relationships between general ideas, and then
fix rules according to which a finite number of pawns moving on a chessboard exhaust a
number of combinations that may even be infinite” (Uses of Literature 39).
Within the story line of Città, the game of chess promptly plays its part. By the
end of the first chapter, the concept of chess is already present. Marco Polo, in order to
communicate, to tell his travel stories, to describe the cities to Kublai Khan, confirms his
inventiveness, even organizing and re-arranging the objects brought from his travels as
pieces in a game of chess; that is, they become a simile for chessmen: “Newly arrived
and totally ignorant of the Levantine languages, Marco Polo could express himself only
with gestures, leaps, cries of wonder and of horror, animal barking or hooting, or with
objects he took from his plumes, pea-shooters, quartzes, which he arranged in front of
him like chessmen” (21).185
The image of the game of chess, besides being the central theme in the opening
and closing dialogues of the eighth chapter, also emerges within the city of Eutropia
found in the fourth chapter (in which the category of “Cities and Signs” ends and that of
“Cities and Names” appear, thus trading or exchanging places, as the disappearance of
185
“Nuovo arrivato e affatto ignaro delle lingue del Levante, Marco Polo non poteva esprimersi altrimenti
che con gesti, salti, grida di meraviglia e d’orrore, latrati o chiurli d’animali, o con oggetti che andava
estraendo dalle sue bisacce: piume di struzzo, cerbottane, quarzi, e disponendo davanti a sé come pezzi
degli scacchi” (21).
127
one allows room for the other). Eutropia, in fact, is the third and thus central to the fifth
category, that of “Trading Cities” (“Le cità e gli scambi” 3). Eutropia means good trade,
change or exchange.
This city, in turn, is composed by multiple cities. “Eutropia is not one, but all
these cities together; only one is inhabited at a time, the others are empty; and this
process is carried out in rotation.” 64). (“Eutropia è non una ma tutte queste città insieme;
una sola è abitata, le altre vuote; e questo si fa a turno” 64). This singular character of the
city seems to provide a varied life to those who live in it.
Nonetheless, the usual turning point immediately follows. Polo relates how the
lives of Eusapia’s inhabitants do not change: the people become bored, telling the same
jokes (simply switching accents), to a humorous point in which, “they open alternate
mouths in identical yawns” (65). (“... ridicono le stesse battute con accenti variamente
combinati; spalancano bocche alternate in uguali sbadigli” 65).
In this city, the inhabitants merely exchange places, moving around, as if they
were in vacant board of chess: “Così la città ripete la sua vita uguale spostandosi in su e
in giù sulla sua scacchiera vuota” (64). (“Thus the city repeats its life, identical, shifting
up and down on its empty chessboard” 64). Peculiarly, in contrast to the horizontal
movements of a game of chess, here, the motion is described as vertical, “in su e in giù”
(“up and down”). Not only does this particularity add a third dimension to the bidimensional chessboard, but also this verticality in movement becomes part of the images
of lightness within the book.
In the eighth chapter, we find how Kublai Khan realizes that the majolica floor
128
resembles a chessboard. Accordingly, he tries to use the rules of the game of chess in
order to decipher Marco’s city stories.
Kublai was a keen chess player; following Marco's movements, he
observed that certain pieces implied or excluded the vicinity of other
pieces and were shifted along certain lines. Ignoring the objects' variety of
form, he could grasp the system of arranging one with respect to the others
on the majolica floor. He thought: "If each city is like a game of chess, the
day when I have learned the rules, I shall finally possess my empire, even
if I shall never succeed in knowing all the cities it contains (122).186
Once having drawn the analogy between each city and a game of chess, the
Emperor concludes that he can figure out all the rules in order to embrace the ownership
and control his Empire, even without the complete knowledge and understanding the
cities. Consequently, he decides in favor of playing chess games rather than sending
Marco Polo to far away lands: “Ormai Kublai Kan non aveva più bisogno di mandare
Marco Polo in spedizioni lontane: lo tratteneva a giocare interminabili partite a scacchi”
(122). “Now Kublai Khan no longer had to send Marco Polo on distant expeditions: he
kept him playing endless games of chess”: “…bastava una scacchiera coi suoi pezzi dalle
forme esattamente classificabili”. (“Actually, it was useless for Marco's speeches to
employ all this bric-a-brac: a chessboard would have sufficed, with its specific pieces”
121).
The game of chess supposes multiple binary oppositions. The chessboard is
composed of (usually) black and white squares, its pieces - kings, queens, bishops,
pawns, towers - are also of two different colors, they move in a two dimensional plane
186
“Kublai era un attento giocatore di scacchi; seguendo i gesti di Marco osservava che certi pezzi
implicavano o escludevano la vicinanza d’altri pezzi e si spostavano secondo certe linee. Trascurando la
varietà di forme degli oggetti, ne definiva il modo di disporsi gli uni rispetto agli altri sul pavimento di
maiolica. Pensò: ”Se ogni città è come una partita a scacchi, il giorno in cui arriverò a conoscerne le regole
possiederò finalmente il mio impero, anche se mai riuscirò a conoscere tutte le città che contiene” (122).
129
(the chessboard), two players are engaged, the game involves visible and invisible
maneuvers, those mentally calculated and those actually executed, and the mathematical
strategies implicate pursuit and retreat. Out of this binary system and its potential
movements in space, its multiple viewpoints, Calvino chooses to create an image of
lightness: “Marco recreated the perspectives and the spaces of black and white cities on
moonlit nights”.
Tornando dalla sua ultima missione Marco Polo trovò il Kan che lo
attendeva seduto davanti a una scacchiera. Con un gesto lo invitò a sedersi
di fronte a lui e a descrivergli col solo aiuto degli scacchi le città che
aveva visitato. Il veneziano non si perse d’animo. Gli scacchi del Gran
Kan erano grandi pezzi d’avorio levigato: disponendo sulla scacchiera
torri incombenti e cavalli ombrosi, addensando sciami di pedine,
tracciando viali diritti o obliqui come l’incedere della regina, Marco
ricreava le prospettive e gli spazi di città bianche e nere nelle notti di luna
(122).187
On the other hand, by looking at its essence, Kublai Khan is able to visualize the
invisible. He begins to perceive the hidden processes and cycles. “Contemplating these
essential landscapes, Kublai reflected on the invisible order that sustains cities, on the
rules that decreed how they rise, take shape and prosper, adapting themselves to the
seasons, and then how they sadden and fall in ruins”(122).188 Kublai Khan could not find
a coherent system or model that would approximate unity and multiplicity like the game
187
“Marco Polo found the Khan awaiting him, seated at a chessboard. With a gesture he invited the
Venetian to sit opposite him and describe, with the help only of the chessmen, the cities he had visited.
Marco did not lose heart. The Great Khan's chessmen were huge pieces of polished ivory: arranging on the
board looming rooks and sulky knights, assembling swarms of pawns, drawing straight or oblique avenues
like a queen's progress, Marco recreated the perspectives and the spaces of black and white cities on
moonlit nights” (122).
188
“Al contemplarne questi paesaggi essenziali, Kublai rifletteva sull’ordine invisibile che regge le città,
sulle regole cui risponde il loro sorgere e prender forma e prosperare e adattarsi alle stagioni e intristire e
cadere in rovina” (122).
130
of chess.189 The process of abstraction leads the Khan to capture the essence. Yet his
quest, when carried to the extreme, was in vain; the goal persisted to be elusive,
“involucri illusori”, which lead to nothing but the wooden chessboard.190 “By
disembodying his conquests to reduce them to the essential,” the Khan arrives nowhere
but back to square one (123, 131). “A forza di scorporare le sue conquiste per ridurle
all’essenza, Kublai era arrivato all’operazione estrema: la conquista definitiva, di cui i
multiformi tesori dell’impero non erano che involucri illusori, si riduceva a un tassello di
legno piallato” (133).
The same phrase appears both at the opening and the closure of the chapter.
Through repetition, redundancy accentuates the Khan’s frustration. “Il Gran Kan cercava
d’immedesimarsi nel gioco: ma adesso era il perché del gioco a sfuggirgli.” (”The Great
Khan tried to concentrate on the game: but now it was the game's purpose that eluded
him” 123, 131).191
As mentioned, Polo responds with his explosive proliferation of storytelling: “The
quantity of things that could he read in a little piece of smooth and empty wood
overwhelmed Kublai; Polo was already talking about ebony forests, about rafts laden
189
“Alle volte gli sembrava d’essere sul punto di scoprire un sistema coerente e armonioso che sottostava
alle infinite difformità e disarmonie, ma nessun modello reggeva il confronto con quello del gioco degli
scacchi” (122).
190
However, it should be noted that the term “involucri illusori”, (or “illusory envelopes” in Cities 133)
suggests the possibility of an exit; a possibility that Polo grasps as he begins to tell stories about “ebony
forests…”
191
Regarding repetition, some passages within the book explicitly reveal the function of redundancy. In the
description of Zirma, (“Cities and Signs 2”) Marco relates: “The city is redundant, it repeats itself so that
something will remain in the mind” (19). (“La città è ridondante: si ripete perché qualcosa arrivi a fissarsi
nella mente”). By the end of this description, he explains “Memory is redundant: it repeats itself so that the
city can begin to exist.” (“ La memoria è ridondante: ripete i segni perché la città cominci a esistere” 19).
131
with logs that come down the rivers, of docks, of women at the windows…” (132).192 It
should be noted that Polo’s creativity does not derive from nothingness but from
preexisting elements, as in any combinatorial system. And as such, it is the element of
surprise, what escapes the system that becomes essential. Marco begins telling other
stories. Again, this is also in accordance to Gödel’s Incompleteness Theorem, as well as
Calvino’s concept of narrative as ars combinatoria.
This passage based on the image of the game of chess, even if just the chessboard,
emphasizes the unlimited possibilities of narrative as a combinatorial game. As
mentioned earlier, the book, like the chessboard, consists of sixty-four parts: fifty-five
cities plus nine chapters. Analogously, the chessboard is divided in sixty-four squares.
The story or legend of chess has many versions, but they all typically chart the
same idea of exemplifying the potential of the game by exposing the concept of
exponential growth, which constitutes an enormously potent notion. In order to facilitate
its comprehension one could use the example of an ancient legend, which, as claimed,
started after a game of chess:
Long ago, a king who was a lively chess player enjoyed challenging his visitors to
play a chess game. On a particular day, a traveling man arrived and the king, in order to
persuade him, offered any prize. (In other versions, when the creator of the game of chess
showed his invention to the ruler of the country, the ruler was so pleased that he gave the
inventor the right to name his prize for the invention).
192
“La quantità di cose che si potevano leggere in un pezzetto di legno liscio e vuoto sommergeva Kublai;
già Polo era venuto a parlare dei boschi d’ebano, delle zattere di tronchi che discendono i fiumi, degli
approdi, delle donne alle finestre ... ” (133-134). 132
The sagacious traveler requested only some grains of rice in this fashion: placing
a single grain on the first square and doubling it on every consequent one, until filling up
the chessboard. He asked the king that for the first square of the chess board he would
receive one grain of wheat (in some telling, rice), two for the second one, four on the
third one, and so forth, doubling the amount each time. The ruler, arithmetically unaware,
quickly accepted the offer.193
Once he had lost the game, the king kept his promise: he ordered one bag of rice
to be brought to the chessboard. He then began putting rice grains in agreement to the
prearrangement: one grain on the first square, two on the second, four on the third, eight
on the fourth and so on, until the sixty fourth square.
The king then discovered that he could never satisfy his promise: it would be
impossible for him to provide enough grains to give a chessboard's worth of grains. (In an
alternate version, when the treasurer spent weeks calculating the amount, the ruler asked
him for the reason for his delay. As the king received the result of the calculation, he was
convinced that it would take more than all the assets of the kingdom to give the inventor
the reward.)
193
Simple addition gives a solution: step by step, one just doubles and adds each term of the series. If the
number of grains is doubled on each consecutive square of the chessboard, then the sum of grains is:
1 + 2 + 4 + 8..., and so forth until the sixty-fourth square, which would contain 9,233,372,036,854,775,808
grains. The total equals an amount much larger than what would usually be expected through mere intuition:
18,446,744,073,709,551,615.
133
`
Due to exponential growth, by the end of the first row, there were 128 grains: by the end
of the second, 32,768 grains…On the twentieth square there would be 1,000,000 grains;
on the fortieth square, 1,000,000,000. Finally on the last square, the sixty-fourth, the
amount would have been more than 18,000,000,000,000,000,000 grains, which is equal
to about 210 billion tons. At ten grains of rice per square inch, the above amount requires
rice fields covering twice the surface area of the Earth, oceans included.194
Stories regarding the chess origin vary; however, they all encompass the same
essential mathematical challenge. The wheat and chessboard problem (sometimes
expressed in terms of rice) can be expressed mathematically in the simple form of a word
problem. If grains were to be placed on a chessboard so that one grain was assigned on
the first square, two on the second, four on the third, and so on (doubling the amount of
grains on each succeeding square), how many grains would be accumulated on the
chessboard at the end?
The calculations for this problem can be used to understand and visualize the
134
quick growth demonstrated by the concepts of exponential growth and geometric
progression. A geometric progression is a sequence of numbers where successive
numbers differ by a constant multiplier, for example, 2, 4, 8, 16 … (Mathematics Harper
Collins Dictionary 245)195. This series may be expressed in various ways. Using
exponents, the series becomes: 20 + 21 + 22 + 23..., and so forth up to 263. The base of
each term, the number two, expresses the doubling at each square, while the exponents
represent the position of each square (zero for the first square, one for the second, etc.).
Chess demonstrates both its immense potential and unchanging rules.
Notwithstanding its rigid constraints, the chessboard grants a perfect example of the
concept of exponential growth.
With respect to exponential expressions, Calvino comments on Jorge Luis Borges
writing. “Nasce con Borges una letteratura elevata al quadrato e nello stesso tempo una
letteratura come estrazione della radice quadrata di se stessa: una "letteratura potenziale",
per usare un termine che sarà applicato più tardi in Francia” (Lezioni americane 58).
Similarly, the exponential aspect of the chessboard, the game of chess exemplifies
the potential of ars combinatoria. The number of possible moves and their combination is
apparently inexhaustible. From this rigid eight by eight grid, thirty-two pieces and set of
rules, the number of potential ways to play a different game, to give it a different form is
theoretically endless.
If the chessboard is static, regardless of its potential to generate exponential
growth, the game, in contrast, is engendered from movement. It is motion, which makes
195
These are different to arithmetic progressions where each number in sequence differs by a constant
amount (the common difference) for example, 3, 6, 9, 12, …, that is, by adding or subtracting.
135
the game possible. But motion requires empty spaces196 or squares for any movement
requires or is defined by a change in location (which is relative, of course, as it requires a
point of reference).
This moving through spaces consists in maneuvers of pursuit and evasion,
creating arrangements, then undoing them, going forward then regressing. Patterns,
which generate with these constant moves, require mathematical agility, in order to see
them before it is no longer possible, as they fleet with the lightness of abstractions.
A
B
C
D
E
F
G
H
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
A
B
C
D
E
F
G
H
3.2 Spaces in Motion: Complex Models
196
Some critics say that in Città this empty space is the role of Bauci, but according to Kublai Khan there
were spaces within cities that were in fact what attracted him the most. Later this issue will be addressed
and discussed in relation to the city of Cecilia.
136
As stated, another mathematical relationship, which shares the duplicity running
throughout the book, is that which can be established with a model. What begins as a
binary association evolves into a multiplicity of intricate correlations, spatial
interconnections pursuing a form.
In Zenobia (“Thin cities” 2), no recollection remains concerning the reasons “to
give their city this form”. The supposition is that it evolved from consecutive
‘superimpositions” rendering its original design impenetrable. But when any inhabitant
visualizes a happy city,” it is always a city like Zenobia that he imagines, with its pilings
and its suspended stairways, a Zenobia perhaps quite different, aflutter with banners and
ribbons, but always derived by combining elements of that first model.” (35)
Quale bisogno o comandamento o desiderio abbia spinto i fondatori di
Zenobia a dare questa forma alla loro città, non si ricorda, e perciò non si
può dire se esso sia stato soddisfatto dalla città quale noi oggi la vediamo,
cresciuta forse per sovrapposizioni successive dal primo e ormai
indecifrabile disegno. Ma quel che è certo è che chi abita a Zenobia e gli si
chiede di descrivere come lui vedrebbe la vita felice, è sempre una città
come Zenobia che egli immagina, con le sue palafitte e le sue scale
sospese, una Zenobia forse tutta diversa, sventolante di stendardi e di
nastri, ma ricavata sempre combinando elementi di quel primo modello
(34).197
Similarly, Clarice, a magnificent city, shows a constant relationship to its
“unequaled model, even with its anguished history: “Più volte decadde e rifiorì, sempre
tenendo la prima Clarice come modello ineguagliabile d’ogni splendore, al cui confronto
lo stato presente della città non manca di suscitare nuovi sospiri a ogni volgere di stelle”
197
“No one remembers what need or command or desire drove Zenobia's founders to give their city this
form, and so there is no telling whether it was satisfied by the city as we see it today, which has perhaps
grown through successive superimpositions from the first, now undecipherable plan. But what is certain is
that if you ask an inhabitant of Zenobia to describe his vision of a happy life, it is always a city like
Zenobia that he imagines, with its pilings and its suspended stairways, a Zenobia perhaps quite different,
aflutter with banners and ribbons, but always derived by combining elements of that first model” (35).
137
(106).198 Yet, even if the relationship to its incomparable, unsurpassed model is slightly
different, its survival form is nothing but a reshuffle of its previous city, a suitable shape
derived from varied combinations:“Messa su coi pezzi scompagnati della Clarice
inservibile, prendeva forma una Clarice della sopravvivenza, tutta tuguri e catapecchie,
rigagnoli infetti, gabbie di conigli. Eppure, dell’antico splendore di Clarice non s’era
perso quasi nulla, era tutto lì, disposto solamente in un ordine diverso ma appropriato alle
esigenze degli abitanti non meno di prima” (106-107).199
Once more the combinatorial process provides the only certainty. Regarding the
essence of the city, precisely, “a given number of objects is shifted within a given space”
and “the rule is to shuffle them each time, and then try to assemble them. “Di sicuro si sa
solo questo: un certo numero d’oggetti si sposta in un certo spazio, ora sommerso da una
quantità d’oggetti nuovi, ora consumandosi senza ricambio; la regola è mescolarli ogni
volta e riprovare a metterli insieme. Forse Clarice è sempre stata solo un tramestio di
carabattole sbrecciate, male assortite, fuori uso”. (108)200
In his ceaseless attempt to portray and expose for Kublai Khan the intricacies of
his Empire, Polo clarifies that all the cities are based on a first model embedded in every
198
“Several times it decayed, then burgeoned again, always keeping the first Clarice as an unparalleled
model of every splendor, compared to which the city's present state can only cause more sighs at every
fading of the stars” (106).
199 “Put
together with odd bits of the useless Clarice, a survivors' Clarice was taking shape, all huts and
hovels, festering sewers, rabbit cages. And yet, almost nothing was lost of Clarice's former splendor; it was
all there, merely arranged in a different order, no less appropriate to the inhabitants' needs than it had been
before” (106).
200
“Only this is known for sure: a given number of objects is shifted within a given space, at times
submerged by a quantity of new objects, at times worn out and not replaced; the rule is to shuffle them each
time, then try to assemble them. Perhaps Clarice has always been only a confusion of chipped gimcracks,
ill assorted, obsolete” (108).
138
city. This is Marco’s native metropolis, Venice. Among Calvino’s numerous essays, one
stands out, not for being particularly well known, but because two years after Città, it
refers to Venice as an “archetype”: “La forza con cui Venezia agisce sulla
immaginazione è quella d’un archetipo vivente che si affaccia sulla utopia” (Saggi II:
2692).
An archetype is the original pattern on which all things of the same kind are
based. In other words, it is a model after which other similar objects are planned,
calculated or designed.201 And yet, models may also trigger further combinations as
elements tend to differ, diverge, vary.
In the introductory dialogue of Città’s sixth chapter, Marco reveals his model.
Each and every city, Polo asserts, is in someway, interconnected to Venice. In their
dialogue, the Khan objects:
"There is still one of which you never speak."
Marco Polo bowed his head.
"Venice," the Khan said. Marco smiled. "What else do you believe I have
been talking to you about?"
The emperor did not tum a hair. "And yet I have never heard you mention
that name.”
And Polo said: "Every time I describe a city I am saying something about
201
Kerstin Pilz quotes David Lock, “The scientist’s primary mode of representation, model making, has
been linked… to the poet’s use of metaphor. In both cases a presumed reality, too tenuous, too complex,
too strange to be represented directly is represented by something else…To delineate the ineffable, the poet
and the scientist alike can only metaphorize “(Science as writing 155; Mapping Complexity 82).
139
Venice" (86).202
This passage resonates throughout the book. In the city of Irene, where different
perspectives design different cities, and “each deserves a different name” Marco unveils:
“…perhaps I have already spoken of Irene under other names; perhaps I have spoken
only of Irene” (125). (“…forse di Irene ho già parlato sotto altri nomi; forse non ho
parlato che di Irene” 125).
Venice, as a model, comprises Marco Polo’s point of reference. “Per distinguere
le qualità delle altre devo partire da una prima città. Per me è Venezia” (RR II: 412).203 In
Città, Venice exists, either as an inconspicuously reminiscence, or as an unequivocal
pattern, within the scheme of each city. “Ogni volta che descrivo una città dico qualcosa
di Venezia” (RR II: 432).204 This being recognized, the model city, Venice, induces a
multiplicity of potential connections, and variations, which ironically engenders
continuousness among the divergent range of cities.
It is no coincidence then, that the city following these statements, the first within
this sixth chapter (and the last of the “trading cities”), should be that of Smeraldina. This
202
“Ne resta una di cui non parli mai.” Marco Polo chinò il capo.
- Venezia, – disse il Kan.
- Marco sorrise. – E di che altro credevi che ti parlassi?
- L’imperatore non batté ciglio. – Eppure non ti ho mai sentito fare il suo nome.
- E Polo: – Ogni volta che descrivo una città dico qualcosa di Venezia.” (86)
203
“To distinguish the other cities’ qualities, I must speak of a first city that remains implicit. For me it is
Venice”(86).
204
“Every time that I describe a city I am saying something about Venice” (86).
Modena considers Smeraldina “the open text par excellence and may be read as the novel writ small” (115).
140
aquatic city appears right after the discussion regarding Venice between Marco and
Kublai Khan (followed by Fillide): “A Smeraldina, città acquatica, un reticolo di canali e
un reticolo di strade si sovrappongono e s’intersecano” (RR II: 433).205
In the city immediately following Smeraldina, namely Fillide, we continue to find
undeniable traces of Venice, from the multiple bridges and canals to the Moorish
windows:”Giunto a Fillide, ti compiaci d'osservare quanti ponti diversi uno dall'altro
attraversano i canali: ponti a schiena d'asino, coperti, su pilastri, su barche, sospesi, con i
parapetti traforati; quante varietà di finestre s'affacciano sulle vie: a bifora, moresche,
lanceolate, a sesto acuto, sormontate da lunette o da rosoni; quante specie di pavimenti
coprano il suolo: a ciottoli, a lastroni, d'imbrecciata, a piastrelle bianche e blu“ (89).206
According to Euclidean geometry, the shortest distance is always a straight line
between two points. This, however, is not the case for the anti-Euclidean Smeraldina:
“Per andare da un posto a un altro hai sempre la scelta fra il percorso terrestre e quello in
barca: e poiché la linea più breve tra due punti a Smeraldina non è una retta ma uno
zigzag che si ramifica in tortuose varianti, le vie che s’aprono a ogni passante non sono
soltanto due ma molte, e ancora alternano per chi alterna traghetti in barca e trasbordi
all’asciutto.” (RR II: 433)207
205
“In Esmeralda, city of water, a network of canals and a network of streets span and intersect each
other.”(88)
206
“When you have arrived at Phyllis, you rejoice in observing all the bridges over the canals, each
different from the others: cambered, covered, on pillars, on barges, suspended, with tracery balustrades.
And what a variety of windows looks down on the streets: mullioned, Moorish, lancet, pointed, surmounted
by lunettes or stained-glass roses; how many kinds of pavement cover the ground: cobbles, slabs, gravel,
blue and white tiles“(89).
207
“To go from one place to another you have always the choice between land and boat: and since the
shortest distance between two points in Esmeralda is not a straight line but a zigzag that ramifies in
tortuous optional routes, the ways that open to each passerby are never two, but many, and they increase
further for those who alternate a stretch by boat with one on dry land” (88).
141
These zigzag movements, these ramifications, these meanderings also describe
the motion within the narrative of Città. In fact, Letizia Modena describes Esmeralda as
“the open text par excellence and may be read as the novel writ small” (115).208
In a fluid world, Euclidean geometry would not make sense. Esmeralda, the antiEuclidean model-city is echoed in the article cited above, “Venezia: archetipo e utopia
della città acquatica”. “La linea più breve che unisce due punti non è mai la linea retta,
tranne che nelle astratte costruzioni di Euclide. Venezia, prima città anti-euclidea, è per
questo il modello di città che ha davanti a sé più avvenire. Prima di tutto, il concetto di
linea più breve tra due punti è relativo: esso varia a seconda di quale moto e quale corpo
traccia il percorso tra i due punti” (Saggi II: 2688).
Moreover, Venice’s net structure is multilevel, creating alternative relationships
between space and movement: “Stabilendo che le vie dei veicoli e quelle dei pedoni non
coincidono mai, Venezia ha fatto di questa relatività dello spazio dal movimento il suo
principio fondamentale (Saggi II: 2688).209
In “Gli dei della città”, Calvino refers to this “intricate and fluid" mapping as
essential for the construction of a city: “ E con occhi nuovi che oggi ci si pone a guardare
la città e ci si trova una città diversa…sono elementi che si compongono in una mappa
208
Referring to Lucia Re (“Textos” 105), Modena adds, “It is not by chance that a prominent Calvino
scholar recognized that Esmeralda conjures up Venice” (116).
209
In Palomar, we find the protagonist in “Il modello dei modelli” going from very rigid modeling to
preferring a more fluid state. In fact, in “Le corse delle giraffe”, we find Mr. Palomar observing the design
created by the discontinuous patterns of motion: “Il signor Palomar, continuando a osservare le giraffe in
corsa, si rende conto d'una complicata armonia che comanda quel trepestio disarmonico, d'una proporzione
interna che lega tra loro le più vistose sproporzioni anatomiche, d'una grazia naturale che vien fuori da
quelle movenze sgraziate. L'elemento unificatore è dato dalle macchie del pelo, disposte in figure irregolari
ma omogenee, dai contorni netti e angolosi; esse si accordano come un esatto equivalente grafico ai
movimenti segmentati dell'animale. Più che di macchie si dovrebbe parlare d'un manto nero la cui
uniformità è spezzata da nervature chiare che s'aprono seguendo un disegno a losanghe: una discontinuità
di pigmentazione che già annuncia la discontinuità dei movimenti” (RR II: 940).
142
intricata e fluida difficile a condurre all’essenzialità d’uno schema. Ma è di qui che
bisogna partire per capire – primo – come la città è fatta, e – secondo – come la si può
rifare” (Saggi I: 349).210
Within Città, this quality grants the chance of multiple routes for Smeraldina’s
inhabitants, to go from place to place, to create new itineraries by “combining elements
of the various routes”. “Combinando segmenti dei diversi tragitti sopraelevati o in
superficie, ogni abitante si dà ogni giorno lo svago d’un nuovo itinerario per andare negli
stessi luoghi. Le vite più abitudinarie e tranquille a Smeraldina trascorrono senza
ripetersi” (89).211 Marco Polo ironically comments how not one person in this city is ever
bored in this multilevel variation.“Così la noia a percorrere ogni giorno le stesse strade è
risparmiata agli abitanti di Smeraldina. E non è tutto: la rete dei passaggi non è disposta
su un solo strato, ma segue un saliscendi di scalette, ballatoi, ponti a schiena d’asino, vie
pensili”(88).212
As usual, in Città the most critical instant in the city’s report, the spinning point,
generates a countermovement where different features, the omissions, the exclusions, the
210
Regarding this inability of Euclidean geometry to delineate some irregularities of forms and patterns,
Benoit Mandelbrot, the inventor of fractals, comments:
“I contend that many of the patterns of Nature are so irregular and fragmentary compared to
Euclidean geometry, show not only a higher degree, but a very different level of complexity. The number
of different scales extension is natural patterns practically infinite. The existence of these patterns
challenges us to study those forms that Euclidean geometry set aside for want of form, to investigate, as it
were, the morphology of "amorphous".
Responding to this challenge, I have designed and developed a new geometry of nature, the use of
which can be implemented in various fields. It describes many of the irregular patterns around us,
identifying a family of shapes that I will call fractals”.
211
“Combining segments of the various routes, elevated or on ground level, each inhabitant can enjoy every
day the pleasure of a new itinerary to reach the same places. The most fixed and calm lives in Esmeralda
are spent without any repetition” (89).
212
“And so Esmeralda's inhabitants are spared the boredom of following the same streets every day. And
that is not all: the network of routes is not arranged on one level, but follows instead an up-and-down
course of steps, landings, cambered bridges, hanging streets” (88).
143
exceptions transpire: “A maggiori costrizioni sono esposte, qui come altrove, le vite
segrete e avventurose” (88).213
I gatti di Smeraldina, i ladri, gli amanti clandestini si spostano per vie più
alte e discontinue, saltando da un tetto all’altro, calandosi da un’altana a
un verone, contornando grondaie con passo da funamboli. Più in basso, i
topi corrono nel buio delle cloache l’uno dietro la coda dell’altro insieme
ai congiurati e ai contrabbandieri: fanno capolino da tombini e da
chiaviche, svicolano per intercapedini e chiassuoli, trascinano da un
nascondiglio all’altro croste di formaggio, mercanzie proibite, barili di
polvere da sparo, attraversano la compattezza della città traforata dalla
raggera dei cunicoli sotterranei (88).214
In Marcovaldo’s story “Il giardino dei gatti ostinati”, the city’s cats stealthily survive as
clandestine inhabitants, hidden within an alternate city. “The city of the cats and the city
of the people lay one within the other, but they are not the same city” (101). “La città dei
gatti e la città degli uomini stanno l'una dentro l'altra, ma non sono la medesima città.
Pochi gatti ricordano il tempo in cui non c'era differenza: le strade e le piazze degli
uomini erano anche strade e piazze dei gatti, e i prati, e i cortili, e i balconi, e le fontane:
si viveva in uno spazio largo e vario”(RR I: 1163).
Within the web’s intervals, they manage to populate a vertical counter-city:
Ma in questa città verticale, in questa città compressa dove tutti i vuoti
tendono a riempirsi e ogni blocco di cemento a compenetrarsi con altri
blocchi di cemento, si apre una specie di controcittà, di città negativa, che
consiste di fette vuote tra muro e muro, di distanze minime prescritte dal
regolamento edilizio tra due costruzioni, tra retro e retro di due costruzioni;
è una città di intercapedini, pozzi di luce, canali d'aerazione, passaggi
213
214
“Secret and adventurous lives, here as elsewhere, are subject to greater restrictions” (88).
“Esmeralda’s cats, thieves, illicit lovers move along higher, discontinuous ways, dropping from a
rooftop to a balcony following gutterings with acrobats' steps. Below, the rats run in the darkness of the
sewers, one behind the other's tail, along with conspirators and smugglers: they peep out of manholes and
drainpipes, they slip through double bottoms and ditches, from one hiding place to another they drag crusts
of cheese, contraband goods, kegs of gunpowder, crossing the city's compactness pierced by the spokes of
underground passages” (88).
144
carrabili, piazzole interne, accessi agli scantinati, come una rete di canali
secchi su un pianeta d'intonaco e catrame, ed è attraverso questa rete che
rasente i muri corre ancora l'antico popolo dei gatti (RR I: 163-164).215
With respect to this verticality, from the first chapter of Città, Isaura, the first of
the “Thin cities”, is described as “a city, which moves entirely upwards” (20).
“…negli archi sottili degli acquedotti, in tutte le colonne d’acqua, i tubi verticali, i
saliscendi, i troppopieni, su fino alle girandole che sormontano le aeree impalcature
d’Isaura, città che si muove tutta verso l’alto” (20). This vertical motion becomes a trait
in the construction of form for the lighter cities, a third dimension contrasting purely
horizontal growth (the bi-dimensional page where the writing lies) and simultaneously
creating an image opposing gravity.
Furthermore, within Smeraldina’s description, the concept of mapping explicitly
reappears: “Una mappa di Smeraldina dovrebbe comprendere, segnati in inchiostro di
diverso colore, tutti questi tracciati, solidi e liquidi, palesi e nascosti” (90). (“A map of
Esmeralda should include, marked in different colored inks, all these routes, solid and
liquid, evident and hidden”) (90). The technique of discerning threads by colors is
proposed in other cities, such as that of Venice described in his essay:
Per sottolineare questa regola, - come su una mappa i due diversi tipi di
via sarebbero segnati in colori differenti , -Venezia caratterizza le vie dei
veicoli come vie acquatiche distinguendole dalle vie terrestri dei pedoni;
cioè sovrappone due reticoli uno solido e l’altro liquido, componendo
tracciati che possono combinarsi e permutarsi in un vario modo
215
“But in this vertical city, in this compressed city where all the voids tend to get filled and every block of
cement rubs into other blocks of cement, there opens a kind of countercity, a negative city, made out of
empty cuts between walls, of minimum distances ruled by the construction regulations between two
buildings, between the back of one building and the back of the next one; it is a city of intervals, skylights,
ventilation ducts, no-parking ways, internal squares, staired access ways, like a web of dry canals on a
planet of plaster and tar, and it is across this web where, against the walls, still races the ancient cat race”
(121).
145
collegando tutti i punti della città nelle due dimensioni acquatica e
terrestre” (Saggi II: 2688).
A dual system of over-imposed networks, one liquid (by water), another solid (by land),
combine and permute the interconnections within all points in the city.
This method of differentiation recurs. In the city of Ersilia, for instance: “A
Ersilia, per stabilire i rapporti che reggono la vita della città, gli abitanti tendono dei fili
tra gli spigoli delle case, bianchi o neri o grigi o bianco–e–neri a seconda se segnano
relazioni di parentela, scambio, autorità, rappresentanza” (76).216
Yet, another characteristic revealed on the essay concerning Venice is developed
in Città mostly, but not exclusively, within the category of “Città continue”. Cities seem
to merge with each other, as their forms become more intricate: “Una cosa Venezia
perderà: il fatto d’essere unica nel suo genere. Il mondo si riempirà di Venezie, ossia di
Supervenezie in cui si sovrapporranno e allacceranno reticoli molteplici a diverse
altezze…” (Saggi II: 2692).
In contrast to the demarcation found in maps and models, traveling through the
cities not only creates a web of fantastic spaces in our minds but also, as differences tend
to disappear, that which seemed to be discrete, discontinuous, like the cities, begins to
show as continuous. An example is the mentioned city of Trude, one of the five
“Continuous Cities”, where this theme is developed. Also “Penthesilea is different”:
You advance for hours and it is not clear to you whether you are already in the
city's midst or still outside it. Like a lake with low shores lost in swamps so
Penthesilea spreads for miles around a soupy city diluted in the plain… And so
you continue, passing from outskirts to outskirts, and the time comes to leave
216
“In Ersilia, to establish the relationships that sustain the city's life, the inhabitants stretch strings from
the corners of the houses, white or black or gray or black-and-white according to whether they mark a
relationship of blood, of trade, authority, agency” (76).
146
Penthesilea…. You have given up trying to understand whether, hidden in some
sac or wrinkle of these dilapidated surroundings there exists a Penthesilea the
visitor can recognize and remember, or whether Penthesilea is only the outskirts
of itself. The question that now begins to gnaw at your mind is more anguished:
outside Penthesilea does an outside exist? Or, no matter how far you go from the
city, will you only pass from one limbo to another, never managing to leave?
(156-158).217
Additionally, Smeraldina’s mapping displays more complexity, in order to
account for spiraling routes: “Più difficile è fissare sulla carta le vie delle rondini, che
tagliano l’aria sopra i tetti, calano lungo parabole invisibili ad ali ferme, scartano per
inghiottire una zanzara, risalgono a spirale rasente un pinnacolo, sovrastano da ogni
punto dei loro sentieri d’aria tutti I punti della città” (90).218
As will be discussed, the structure of the spiral includes images of the continuous
and the hidden, the last two themes in Città. Concerning this part of our study, this twisty
shape is of interest because its winding form revolves around a center.219
3.3 The Center: Lightness and the Spiral
“Ma c’è anche l’atra via, quella che sostiene che il senso di un libro simmetrico va
cercato nel mezzo…” (Città x)
217
“Pentesilea è diverso. Sono ore che avanzi e non ti è chiaro se sei già in mezzo alla città o ancora fuori.
Come un lago dalle rive basse che si perde in acquitrini, così Pentesilea si spande per miglia intorno in una
zuppa di città diluita nella pianura…Così prosegui, passando da una periferia all’altra, e viene l’ora di
partire da Pentesilea…Se nascosta in qualche sacca o ruga di questo slabbrato circondario esista una
Pentesilea riconoscibile ericordabile da chi c’è stato, oppure se Pentesilea è solo periferia di se stessa e ha il
suo centro in ogni luogo, hai rinunciato a capirlo. La domanda che adesso comincia a rodere nella tua testa
è più angosciosa: fuori da Pentesilea esiste un fuori? O per quanto ti allontani dalla città non fai che passare
da un limbo all’altro e non arrivi a uscirne?” (156-158).
218
“It is more difficult to fix on the map the routes of the swallows…spiraling upward…dominating from
every point of their airy paths all the points of the city” (90).
219
Martin McLaughlin comments “Of course, Calvino’s interest in complex structures go back at least to
his division of the 1958 collection of Racconti and to the five cycles of Marcovaldo (1963). But we have
noted his growing preoccupation with mathematical problems … and by 1972 he was even more obsessed
with the mathematical problems of his fellow members of the Oulipo…who engrossed in attempts to make
mathematical devices structure literary texts by acting as constraintes. It is these more complex
mathematical constraints that subtend the intricate structure of Invisible Cities” (Italo Calvino 101).
147
A sequence with an odd number of elements, guarantees a central element. Le
città invisibili consists of fifty-five cities divided in nine chapters. Correspondingly, the
book ought to point towards a structural center. The central chapter would be the fifth
one. Its position leaves four chapters behind and four ahead within the book’s sequence:
1 2 3 4 5 6 7 8 9.
Similarly, in this fifth chapter, lies the “central” city, that of Bauci. Within the
fifty-five cities, this city would be the twenty-eighth, leaving twenty-seven cities to each
of its “sides”, before and after it, inside the book’s sequence. Besides, the theme of eyes,
“Le città e gli occhi”, appears as the sixth of eleven themes, thus taking the central
position in the series of themes (1 2 3 4 5 6 7 8 9 10 11). Moreover, it is no wonder, that
the central city of the book is also the third, and thus again central within its category,
formed by the five cities related to the theme of eyes, or under the rubric “Le città e gli
occhi”. Visibility of the invisible appears again as a central aspect to the book.
The five cities on this fifth chapter, referred as filiformi by the author on the
“Presentation” of Città, revolve around the theme of lightness. They are: Octavia (“Le
città sottili 5”); the spider web city, Ersilia (“Scambi 4”), made of threads and
relationships; Bauci (“La città e gli occhi 2”), which contemplates its absence; Leandra
(“La città e il nome 2”), with infinite, minuscule Gods of two species; and Melania (“La
città e i morti 1”) where the same dialogue seems to go on forever, and its inhabitant’s
life is too short to notice change.220
220
“The cities in chapter 5, to which Octavia, Ersilia, Baucis, Leandra, and Melania all belong, embody
lightness more than any others in the novel. Evidence for this claim is to be found in the preface, in which
Calvino limits himself to a single observation about his urban icons from the perspective of the visual, but
he does not use the word visual… This observation offers a hermeneutical key to understanding the novel
as much more than a postmodern experimentation with literary forms, for it places lightness in the
foreground and marries it to the visionary, or the transcendent visual” (Modena 155).
148
Concerning the construction of Le città invisibili, Calvino relegates the text with
its own power: “Il libro, come ho spiegato, si è fatto un po’ da sé, ed è solo il testo com’è
che può autorizzare o escludere questa o quella lettura”(x-xi). Still, the author’s remarks
point to the central chapter, for its attainment of both lightness and visibility: “Come
lettore tra gli altri, posso dire che nel capitolo quinto, che sviluppa nel cuore del libro un
tema di leggerezza stranamente associato a quello della città, ci sono alcuni pezzi che
considero migliori come evidenza visionaria, e forse queste figure più filiformi (“città
sottili” o altre) sono la zona più luminosa221del libro. Non saprei dire di più” (xi).222
Accordingly, the theme of lightness is interconnected together with that of the
city. This strange link that Calvino sees between the cities and lightness, especially the
center, could also be said about mathematics and lightness. Mathematical concepts, as
investigated here, are instrumental to design and visualize these fantastic city spaces in
our minds; spaces, which turn out to be reversible, interchangeable, multipliable and
divisible, due to their combinatorial nature and its pursuit for lightness.
Besides visibility, lightness, and its relationship to mathematics, is of most
importance to this research; it is in fact central to the book as well as to the author. As
one finds mathematical terms and concepts in the text, it becomes increasingly evident,
particularly in these cities to which the author refers to as filiformi, for example, the
“Thin Cities”223, that mathematics are not just useful in the search for exactitude or
221
The word “luminosa” (“luminous”) in this statement draws a relationship between light and lightness,
thus creating a link between visibility and lightness.
222
“As a reader among others, I can say that in Chapter 5, which develops at the very heart of the book a
theme of lightness strangely associated with the theme of the city, there are some passages that I think are
better as visionary evidence, and perhaps these threadlike shapes (the ‘Thin cities’ and others) are the most
luminous area in the entire book” (xi).
149
visibility, but also to remove weight, to achieve lightness and clarity of vision in our
minds. Just like in a mathematical puzzle, mental agility is required, to place the pieces
according to a combinatorial game into a coherent whole, a configuration, which can then
be viewed from different perspectives by way of a map or a chess game, or visualized
like a shape that twists and turns, among others, a spiral.
We will explore, how visual structures, achieved by mathematical relationships of
various kinds, as demonstrated, relate to the theme of lightness; central to this book, and
one of the literary values Calvino wanted to emphasize in his Six Memos for the Next
Millennium (Lezioni americane).
Lightness, the theme and title of first lecture of his collection Lezioni americane,
is central, both thematically and structurally to Le città invisibili. “La leggerezza non è
altro che la vittoria sulla propria pesantezza” (RR III 416). As the author explains, his
forty years of writing were devoted mostly to acquiring weightlessness in the texts and
their structures.“Dopo quarant’anni che scrivo fiction… la mia operazione è stata il più
delle volte una sottrazione di peso; ho cercato di togliere peso ora alle figure umane, ora
ai corpi celesti, ora alle città, soprattutto ho cercato di togliere peso alla struttura del
racconto e al linguaggio” (Lezioni americane 7).224
Calvino reveals his strategy for fighting weight, that is, (for) achieving lightness
within the heaviness of the world in his writing. “Whenever humanity seems condemned
to heaviness, I think I should fly like Perseus to a different space…I have to change my
223
The five cities within this category are all discussed in this study: Isaura, Zenobia, Armilla, Sofronia and
Octavia.
224
“After forty years of writing fiction, my working method has more than not involved the subtraction of
weight. I have tried to remove weight, sometimes from people, sometimes from heavenly bodies,
sometimes from cities; above all I have tried to remove weight from the structure of novels and from
language” (Six Memos 3).
150
approach, look at the world from a different perspective, with a different logic and with
fresh methods of cognition and verification” (Memos 7).225
In order to gain a fresh, enriching yet objective point of view, he confesses how at
times he searched within science: “But if literature is not enough to assure me that I am
not chasing dreams, I look to science to nourish my visions in which all heaviness
disappears” (Six Memos 8). “Nell'universo infinito della letteratura s'aprono sempre altre
vie da esplorare, nuovissime o antichissime, stili e forme che possono cambiare la nostra
immagine del mondo... Ma se la letteratura non basta ad assicurarmi che non sto solo
inseguendo dei sogni, cerco nella scienza alimento per le mie visioni in cui ogni
pesantezza viene dissolta...” (12).226
Calvino affirms that in the 1980’s every scientific field- evidently based on
mathematics- was determined to reveal how creation is sustained by the minutest of units.
“Oggi ogni ramo della scienza sembra ci voglia dimostrare che il mondo si regge su
entità sottilissime: come i messaggi del DNA, gli impulsi dei neuroni, i quarks, i neutrini
225
“Nei momenti in cui il regno dell'umano mi sembra condannato alla pesantezza, penso che dovrei volare
come Perseo in un altro spazio. Non sto parlando di fughe nel sogno o nell'irrazionale. Voglio dire che devo
cambiare il mio approccio, devo guardare il mondo con un'altra ottica, un'altra logica, altri metodi di
conoscenza e di verifica.” (12).
226
Calvino’s interdisciplinary interest, particularly in science and mathematics, has been commented by
well know authorities in these fields. In Mapping Complexities, Kerstin Pilz comments: “At his death
several obituaries paid tribute to Calvino as a uniquely interdisciplinary writer and thinker, confirming his
interest and knowledge of science beyond a merely superficial acquaintance. Massimo Piatelli Palmarini,
the philosopher and personal friend of Calvino, stresses on his obituary the writer’s lifelong interest in
science: “Insaziabile curioso lo era in particolare modo per la scienza. Leggeva avidamente e puntualmente
ogni numero di Scientific American …Trovava che la realtà rivelataci dalle scienze superava speso la
fantasia”. (Piatelli Palmarini, 1985) (Pilz 27) “Primo Levi on his obituary likewise confirmed Calvino’s
interest in science: “aveva fame di scienza, la coltivava, se ne nutriva da dilettante colto e critico, e di essa
nutriva i suoi libri più maturi” (Levi 1985) (Pilz 27).
151
vaganti nello spazio dall'inizio dei tempi [...] (Lezioni 18). 227
All of these examples of subtleties, are not only interconnected to lightness and
combinatorial art, but further demonstrate Calvino’s concern with the latest scientific and
mathematical developments. Among these extraordinary revelations, the DNA is
essential to this research for its twisty, spiral figure and its ability to constantly double
itself. “The combination of elemental figures determining the variety it forms connects
Epicurean science with the genetics of DNA” (“Cyrano and the Moon”. Uses 333).
Even more, Calvino, who unfortunately did not live to see the Internet, anticipates
its principle in numerous ways: “Poi, l'informatica. E' vero che il software non potrebbe
esercitare i poteri della sua leggerezza se non mediante la pesantezza dell’hardware; ma è
il software che comanda, che agisce sul mondo esterno e sulle macchine, le quali esistono
solo in funzione del software, si evolvono in modo d'elaborare programmi sempre più
complessi” (Lezioni 12).228
Several sources to which Calvino recurs in order to connect his line of argument
are interesting to this investigation. He refers to Lucretius’ De Rerum Natura (1st century
BC) as “the first great work of poetry in which knowledge of the world tends to dissolve
the solidity of the world, leading to a perception of all that is infinitely mute, light and
227
“Today every branch of science seems intent on demonstrating that the world is supported by the most
minute entities, such as the messages of DNA, the impulses of neurons, and quarks wondering through
space since the beginning of time” (Memos 8).
228
“Then we have computer science. It is true that software cannot exercise its powers of lightness except
through the weight of hardware. Bit it is software that gives the orders, acting on the outside world and on
machines that exist only as function s of software and evolve so that they can work out ever more complex
programs” (Memos 8).
152
mobile.”(Memos 8)229 Elsewhere, in reference to Cyrano de Bergerac, Calvino maintains,
“His poetic imagination derives from cosmic feelings which led him to evoke the
sentiments of Lucretius atomism” (“Cyrano and the Moon”. Uses 333).
This view is consistent with Calvino’s combinatorial construction of Città, as seen
in the index and explored here while traveling through the cities.230 Lucretius’ ideas are
in accordance to Calvino’s book as a whole, even with respect to invisibility. “Lucretius
set out to write the poem of physical matter, but he warns us at the onset that this matter
is made up of invisible particles...the first thing that he tells us is that emptiness is just as
concrete as solid bodies” (Six Memos 8).231
Moreover, Calvino concludes: “Lucretius chief concern is to prevent the weight of
matter from crushing us” (Six Memos 8-9). “La più grande preoccupazione di Lucrezio
sembra quella di evitare che il peso della materia ci schiacci” (Lezioni 13). Similarly, in
the opening of the central chapter of Città, we encounter Kublai Khan also preoccupied
with this issue of heaviness: that of his Empire’s cities. “The Great Khan contemplates an
empire covered with cities that weigh upon the earth and upon mankind, crammed with
wealth and traffic, over laden with ornaments and offices, complicated with mechanisms
and hierarchies, swollen, tense, ponderous. ‘The empire is being crushed by its own
weight’, Kublai thinks, and in his dreams now cities light as kites appear, pierced cities
229
“Il De rerum natura di Lucrezio è la prima grande opera di poesia in cui la conoscenza del mondo
diventa dissoluzione della compattezza del mondo, percezione di ciò che è infinitamente minuto e mobile e
leggero” (Lezioni americane 13).
230
In the central chapter, for instance, the mentioned city of Leandra is protected by gods of two species:
“Both are too tiny to be seen and too numerous to be counted.” (78)
231
“Lucrezio vuole scrivere il poema della materia ma ci avverte subito che la vera realtà di questa materia
è fatta di corpuscoli invisibili. È il poeta della concretezza fisica, vista nella sua sostanza permanente e
immutabile, ma per prima cosa ci dice che il vuoto è altrettanto concreto che i corpi solidi” (Lezioni 13).
153
like laces, cities transparent as mosquito netting, cities like leaves' veins, cities lined like
a hand's palm, filigree cities to be seen through their opaque and fictitious thickness.”
(73)232
Lightness also constitutes the main feature in Calvino’s “La geografia delle fate”:
“ “Il primo attributo è la leggerezza.” (139). Still, this valued quality reveals
discontinuous, subtlle evasive forms, and only rarely noticed: “La loro apparenza e forse
la loro stessa presenza è discontinua: solo chi è dotato di una seconda vista li può
percepire, e sempre per brevi istanti perché appaiono e scompaiono…chi dice “strane
ragnatele”, ci dice “arcobaleni impalpabili…” (Sabbia 139).
Calvino specifies, in his Lezioni, yet another crucial aspect on Lucretius: how he
launches the concept of clinamen. “Even while laying down the rigorous mechanical laws
that determine every event, he feels the need to allow atoms to make unpredictable
deviations from the straight line, thereby ensuring freedom both to atoms and to human
beings” (9). 233
Clinamen is the Latin name given by Lucretius to the unpredictable swerve of
atoms. During the time the Lucretius lived in Rome, Epicureanism was flourishing in this
city. Epicurean Physics, just like nineteenth century Physics, is based on atomism. By
232
“Il Gran Kan contempla un impero ricoperto di città che pesano sulla terra e sugli uomini, stipato di
ricchezze e d’ingorghi, stracarico d’ornamenti e d’incombenze, complicato di meccanismi e di gerarchie,
gonfio, teso, greve. ‘È il suo stesso peso che sta schiacciando l’impero’, pensa Kublai, e nei suoi sogni ora
appaiono città leggere come aquiloni, città traforate come pizzi, città trasparenti come zanzariere, città
nervatura di foglia, città linea della mano, città filigrana da vedere attraverso il loro opaco e fittizio
spessore”(74).
233
“Al momento di stabilire le rigorose leggi meccaniche che determinano ogni evento, egli sente il
bisogno di permettere agli atomi delle deviazioni imprevedibili dalla linea retta, tali da garantire la libertà
tanto alla materia quanto agli esseri umani” (13).
154
watching matter dividing, subdividing, reuniting into various forms, Lucretius came to
the conclusion that the Universe is made small building blocks called atoms.234
In his reference to Lucretius, as with other authors, Calvino reveals sources of
inspiration. “The atomizing of things extends also to the visible aspects of the world and
it is here that Lucretius is at his best as a poet: the little motes of dust swirling…the
minuscule shells, all similar but each one different, that waves gently cast up on the
bibula arena, the “imbibing sand” or the spider webs that wrap themselves around us
without our noticing them as we walk along” (Six Memos 9).235 In this case the common
elements with Città speak for themselves: the “polviscolare”, the “punctiform”, the spiral,
the spider webs as images of the unseen world, are integral to the invisible cities. 236
Another one of Calvino’s admired poems, written fifty years after Lucretius’ De
Rerum Natura, Ovid’s Metamorphosis (8 AD) provides him enriching examples: “For
Ovid, too, everything can be transformed into something else, and knowledge of the
world means dissolving the solidity of the world” (Six Memos 9). “Anche per Ovidio
tutto può trasformarsi in nuove forme; anche per Ovidio la conoscenza del mondo è
dissoluzione della compattezza del mondo” (14). Again, Calvino’s emphasis is on
234
In his essay “Prose and Anticombinatorics”, Calvino states: “The structures chosen by the author are
relatively few in number, but the possible realizations are combinatorial exponential…the aid of the
computer, far from replacing the creative act of the artist, permits the latter rather to liberate himself from
the slavery of a combinatory search, allowing him the best chance of concentrating on this “clinamen”,
which alone, can make the text a true work of art” (Warren F. Motte 143-152).
235
“Questa polverizzazione della realtà s'estende anche agli aspetti visibili, ed è là che eccelle la qualità
poetica di Lucrezio: i granelli di polvere che turbinano in un raggio di sole in una stanza buia ; le minute
conchiglie tutte simili e tutte diverse che l'onda mollemente spinge sulla bibula harena, sulla sabbia che
s'imbeve le ragnatele che ci avvolgono senza che noi ce ne accorgiamo mentre camminiamo” (Lezioni 13).
236
“For Lucretius, letters were atoms in continual motion, creating the most diverse words and sounds by
means of their permutations. This notion was taken up by a long tradition of thinkers for whom the world’s
secrets were contained in the combinatorial of he signs used in writing: one thinks of the Ars Magna of
Raymo Lull the Cabala of the Spanish rabbis and Pico della Mirandola…Even Galileo saw the alphabet as
the model for all combinations of minimal units…And then Leibniz… “ (Six Memos 26).
155
lightness, on its elusive transformations for which he is searching, to which he dedicates a
whole lecture, and which we find in Città, particularly in the center.
Of most interest to this research (in regards to mathematical and scientific
philosophy) is how Calvino summarizes the views on lightness of these two great
authors: “In both Lucretius and Ovid, lightness is a way of looking at the world based on
philosophy and science: the doctrines of Epicurus for Lucretius and those of Pythagoras
for Ovid” (Memos 10). (“Tanto in Lucrezio quanto in Ovidio la leggerezza è un modo di
vedere il mondo che si fonda sulla filosofia e sulla scienza: le dottrine di Epicuro per
Lucrezio, le dottrine di Pitagora per Ovidio” 14). (Pythagoras, as pointed out, is one of
the mathematicians mentioned in Città).
In order to continue to illustrate what he means by lightness, Calvino continues to
refer to other authors (for example, Cervantes and Boccaccio). After reviewing
Cavalcanti, he comes to a conclusion, which emphasizes his previous point: “This
discussion of Cavalcanti has served to clarify (at least for myself) what I mean by
“lightness”. Lightness for me goes with precision and determination, not with vagueness
and the haphazard” (Six Memos 16). (L'essermi soffermato su Cavalcanti m'è servito a
chiarire meglio (almeno a me stesso) cosa intendo per "leggerezza". La leggerezza per me
si associa con la precisione e la determinazione, non con la vaghezza e l'abbandono al
caso (20).
Here he seems to find an explicit reason for using mathematical sciences in his
search for lightness: precision. But Calvino reinforces the point of his argument by
referring to Henry James similarity with Le città invisibili: “There is the narration of a
train of thought or psychological process in which subtle and imperceptible elements are
156
at work or any kind of description that involves a high degree of abstraction” (Six Memos
17).237 As in Calvino’s Invisible cities “there is a visual image of lightness that acquires
emblematic value” (Six Memos 17).
Likewise, humor is linked to lightness, as Calvino perceives it to be another result
of the subtraction of weight: “As melancholy is sadness that has taken on lightness, so
humor is comedy that has lost its bodily weight” (Six Memos 19). In fact, in allusion to
“As You like It”, Calvino uses the paradoxical term of “humorous sadness” (Six Memos
19-20).
In his quest for illustrations of lightness, Calvino consistently revives his first
Norton lecture. Considering Cyrano de Bergerac as “the first poet of atomism in modern
literature”, he explains how “Cyrano extols the unity of all things, animate or inanimate,
the combinatoria of elementary figures that determine the variety of living forms” (Six
Memos 20).
Seguace del sensismo di Gassendi e dell'astronomia di Copernico, ma
soprattutto nutrito della "filosofia naturale" del Rinascimento italiano Cardano, Bruno, Campanella - Cyrano è il primo poeta dell'atomismo
nelle letterature moderne. In pagine la cui ironia non fa velo a una vera
commozione cosmica, Cyrano celebra l'unità di tutte le cose, inanimate o
animate, la combinatoria di figure elementari che determina la varietà
delle forme viventi, e soprattutto egli rende il senso della precarietà dei
processi che le hanno create: cioè quanto poco è mancato perché l'uomo
non fosse l'uomo, e la vita la vita, e il mondo un mondo ( Lezioni 26).
Following this line of thought, Calvino proclaims Cyrano to be a pioneer:
In my discussion of lightness, Cyrano is bound to figure chiefly because
(before Newton) he felt the problem of universal gravitation. Or rather, it
237
“…la narrazione d'un ragionamento o d'un processo psicologico in cui agiscono elementi sottili e
impercettibili, o qualunque descrizione che comporti un alto grado d'astrazione” (21).
157
is the problem of escaping the force of gravity that so stimulates his
imagination as to lead him to think up a whole series of ways of reaching
the moon, each one more ingenuous than the last…” (Six Memos 22).238
In comparison to Jonathan Swift, who “was a contemporary and adversary of
Newton, Calvino declares Voltaire “an admirer of Newton” as he, ironically “imagined a
giant called Micromègas239, who in contrast to Swift’s giants is defined not by his bulk
but by dimensions expressed in figures, by spatial and temporal properties enumerated in
the rigorous, impassive terms of scientific treatises” (Six Memos 23).
Physically, suspension, or floating in space results from the mathematical
balancing of (opposite) forces. This constitutes an essential principle in the construction
of the lighter images within Le città invisibili. Besides, weight is but a measurement of
the invisible force of gravity240. “One might say that, in Newton’s theories, what most
strikes the literary imagination is not the conditioning of everything and everyone by the
inevitability of its own weight, but rather the balance of forces that enables the heavenly
bodies to float” (Six Memos 23).241 For Calvino, “The eighteenth-century imagination is
full of figures suspended in the air” (Six Memos 23). (“ L'immaginazione del secolo
XVIII è ricca di figure sospese per aria.” Lezioni 30).
238
“Nella mia trattazione sulla leggerezza, Cyrano figura soprattutto per il modo in cui, prima di Newton,
egli ha sentito il problema della gravitazione universale; o meglio, è il problema di sottrarsi alla forza di
gravità che stimola talmente la sua fantasia da fargli inventare tutta una serie di sistemi per salire sulla luna,
uno più ingegnoso dell'altro”(28).
239
Within the decimal system, micro is the prefix used for a specific quantity divided by a million, i.e., a
millionth part, whereas mega denotes millions, that is, an amount multiplied by a million. The name
Micromegas is hence, a mathematical pun.
240
241
In Physics, the unit for measuring weight is the newton, whereas kilograms or pounds are units for mass.
“Si direbbe che nelle teorie di Newton ciò che colpisce l'immaginazione letteraria non sia il
condizionamento d'ogni cosa e persona alla fatalità del proprio peso, bensì l'equilibrio delle forze che
permette ai corpi celesti di librarsi nello spazio” (Lezioni 29-30).
158
The book of Thousand and One Nights, a source of inspiration for Calvino, could
have also influenced the images of his “lighter” cities. “It is no coincidence that at the
beginning of that century Antoine Galland’s French translation of the Thousand and One
Nights opened up the imagination of the West to the Eastern sense of marvel… in this
drive to make the imagination exceed…a constant challenge to the laws of gravity”
(Memos 23). (“Non per nulla agli inizi del secolo la traduzione francese delle Mille e una
Notte di Antoine Galland aveva aperto alla fantasia occidentale gli orizzonti del
meraviglioso orientale” Lezioni 30).
Calvino’s lecture “Lightness” concludes with an enchanting image, which captivates our
attention back to the discussion on lightness in Città:
As soon as the moon appears in poetry, it brings with it sensations of
lightness, suspension, a silent calm enchantment...When I began thinking
about these lectures, I wanted to leave one whole talk to the moon…Then
I decided that the moon should be left entirely to Leopardi. For the
miraculous thing is that he takes the weight out of language so that it
resembles moonlight (Six Memos 24).242
In Città, the moon makes distinctive appearances, not only within but also outside
of the fifty-five cities. The most memorable is Lalage, the one city dreamed by Kublai
Khan 243when he becomes concerned with the “heaviness” of the Empire.244 The emperor
242
“La luna, appena s'affaccia nei versi dei poeti, ha avuto sempre il potere di comunicare una sensazione
di levità, di sospensione, di silenzioso e calmo incantesimo…Poi ho deciso che la luna andava lasciata tutta
a Leopardi. Perché il miracolo di Leopardi è stato di togliere al linguaggio ogni peso fino a farlo
assomigliare alla luce lunare” (31).
“Calvino also points out Leopardi’s early knowledge of Newton: “When he was fifteen years old, Giacomo
Leopardi wrote an amazingly erudite History of Astronomy, in which among other things he sums up
Newton’s theory. (Memos 24) “Giacomo Leopardi a quindici anni scrive una storia dell'astronomia di
straordinaria erudizione, in cui tra l'altro compendia le teorie newtoniane” (Lezioni 30-31).
243
244
Here, one could point out a biblical echo, namely Daniel explaining Pharaoh’s dream (Daniel 2:46-48).
In contrast, Zobeide “well exposed to the moon” and “with streets wound about themselves as in a
skein” ends up becoming a “trap”: “Di là, dopo sei giorni e sette notti, l’uomo arriva a Zobeide, città
bianca, ben esposta alla luna, con vie che girano su se stesse come in un gomitolo” (45).
159
(Kublai Khan), at the beginning of the fifth and central chapter of Città, is found
reflecting on the heaviness of the cities. Precisely at this point, he dreams of Lalage, the
city particularly privileged by the moon:
– Ti racconterò cosa ho sognato stanotte, – dice a Marco. – In mezzo a
una terra piatta e gialla, cosparsa di meteoriti e massi erratici, vedevo di
lontano elevarsi le guglie d’una città dai pinnacoli sottili, fatti in modo
che la Luna nel suo viaggio possa posarsi ora sull’uno ora sull’altro, o
dondolare appesa ai cavi delle gru.
E Polo: – La città che hai sognato è Lalage. Questi inviti alla sosta nel
cielo notturno i suoi abitanti disposero perché la Luna conceda a ogni
cosa nella città di crescere e ricrescere senza fine.
– C’è qualcosa che tu non sai, – aggiunse il Kan. – Riconoscente la Luna
ha dato alla città di Lalage un privilegio più raro: crescere in
leggerezza.245
The image of such buoyant growth transports this discussion back to Lucretius and to
Ovid’s concepts of lightness: “Se il mondo di Lucrezio è fatto d'atomi inalterabili, quello
d'Ovidio è fatto di qualità, d'attributi, di forme che definiscono la diversità d'ogni cosa e
pianta e animale e persona; ma questi non sono che tenui involucri d'una sostanza
comune che, … può trasformarsi in quel che vi è di più diverso. E' nel seguire la
continuità del passaggio da una forma a un'altra che Ovidio dispiega le sue ineguagliabili
doti” (14).
Together with the theme of lightness and sharing its concentric position there is
the spider web images form by the cities. Through a multiplicity of relationships Calvino
245
“I shall tell you what I dreamed last night," he says to Marco. "In the midst of a flat and yellow land,
dotted with meteorites and erratic boulders, I saw from a distance the spires of a city rise, slender pinnacles,
made in such a way that the moon in her journey can rest now on one, now on another, or sway from the
cables of the cranes." And Polo says: "The city of your dream is Lalage. Its inhabitants arranged these
invitations to rest in the night sky so that the moon would grant everything in the city the power to grow
and grow endlessly" "There is something you do not know," the Khan adds. "The grateful moon has
granted the city of Lalage a rarer privilege: to grow in lightness" (74).
160
manages to construct a net image. This structure allows for cities to live, existing not just
in their own space, but interlaced within the book space, connected in our minds. The net
structure, is not only essential to the combinatorial structure of the book, but becomes
explicit in many cities specially, but not exclusively, within the thin cities (sottili) and
those with threadlike forms (filiformi).
At times, the cities appear suspended over a void, as if weightless. Such is the
case of Octavia, the first of this fifth chapter and the fifth and last of the “Thin cities”
(“Città sottili 5”):
Se volete credermi, bene. Ora dirò come è fatta Ottavia, città–ragnatela.
C’è un precipizio in mezzo a due montagne scoscese: la città è sul vuoto,
legata alle due creste con funi e catene e passerelle. Si cammina sulle
traversine di legno, attenti a non mettere il piede negli intervalli, o ci si
aggrappa alle maglie di canapa. Sotto non c’è niente per centinaia e
centinaia di metri: qualche nuvola scorre; s’intravede più in basso il fondo
del burrone (75).246
The net constitutes the city’s base, its foundation. It serves a double purpose: as a
pathway, or as support: everything hangs from it.
Questa è la base della città: una rete che serve da passaggio e da sostegno.
Tutto il resto, invece d’elevarsi sopra, sta appeso sotto: scale di corda,
amache, case fatte a sacco, attaccapanni, terrazzi come navicelle, otri
d’acqua, becchi del gas, girarrosti, cesti appesi a spaghi, montacarichi,
docce, trapezi e anelli per i giochi, teleferiche, lampadari, vasi con piante
dal fogliame pendulo (75).247
246
“If you choose to believe me, good. Now I will tell how Octavia, the spider-web city, is made. There is a
precipice between two steep mountains: the city is over the void, bound to the two crests with ropes and
chains and catwalks. You walk on the little wooden ties, careful not to set your foot in the open spaces, or
you cling to the hempen strands below there is nothing for hundreds and hundreds of feet: a few clouds
glide past; farther down you can glimpse the chasm's bed” (75).
247
“This is the foundation of the city: a net which serves as passage and as support. All the rest, instead of
rising up, is hung below: rope ladders, hammocks, houses made like sacks, clothes hangers, terraces like
gondolas, skins of water, gas jets, spits, baskets on strings, dumb-waiters, showers, trapezes and rings for
children's games, cable cars, chandeliers, pots with trailing plants” (75).
161
In the end, Octavia’s lightness provides one certainty: the existence of this spider-web
city, as suspended from a net, has a limit of time: “Sospesa sull'abisso, la vita degli
abitanti d'Ottavia è meno incerta che in altre città. Sanno che più di tanto la rete non
regge” (75).1
The description of the city of Ersilia concludes, with a similar image to the city of
Octavia. “Così viaggiando nel territorio di Ersilia incontri le rovine delle città
abbandonate, senza le mura che non durano, senza le ossa dei morti che il vento fa
rotolare: ragnatele di rapporti intricati che cercano una forma (76).248
Threads established by, and also manifesting, the relationships between its
inhabitants construct the city. “A Ersilia, per stabilire i rapporti che reggono la vita della
città, gli abitanti tendono dei fili tra gli spigoli delle case, bianchi o neri o grigi o biancoe-neri a seconda se segnano relazioni di parentela, scambio, autorità, rappresentanza”
(76).249 As soon as there is no more room to draw further inter-connections among
themselves, the inhabitants move, relocate their city to another, free space, where they
restart entwining another city-form. These interlaced shapes are the only traces left
behind as the process continues: “Quando i fili sono tanti che non ci si può più passare in
mezzo, gli abitanti vanno via: le case vengono smontate; restano solo i fili e i sostegni dei
fili” (76).250
248
“Thus, when traveling in the territory of Ersilia, you come upon the ruins of the abandoned cities,
without the walls which do not last, without the bones of the dead which the wind rolls away: spider- webs
of intricate relationships seeking a form” (76).
249
“In Ersilia, to establish the relationships that sustain the city's life, the inhabitants stretch strings from
the corners of the houses, white or black or gray or black-and-white according to whether they mark a
relationship of blood, of trade, authority, agency” (76).
250
“When the strings become so numerous that you can no longer pass among them, the inhabitants leave:
162
Only from afar, at a distance, (like Bauci, Irene) can the labyrinthine networks be
appreciated. From this vantage perspective, the city’s shape persists delineated by
threads: “Dalla costa d’un monte, accampati con le masserizie, i profughi di Ersilia
guardano l’intrico di fili tesi e pali che s’innalza nella pianura. È quello ancora la città di
Ersilia, e loro sono niente” (76).251
This city form, this spider-web is in constant movement as the inhabitants
continue to reconstruct their, knitting variations of the same design, then leaving it
behind, to begin again: “Riedificano Ersilia altrove. Tessono con i fili una figura simile
che vorrebbero più complicata e insieme più regolare dell’altra. Poi l’abbandonano e
trasportano ancora più lontano sé e le case” (76).252
These motions of interconnections, these intricate webs - relationships in motion design other forms in space: a focal one in Città is the spiral.
“La spirale gira e segue insieme lo svolgersi della storia nel tempo e l’itinerario
nello spazio, per cui il racconto non ritorna mai negli stessi luoghi” (Sabbia 100).
In his “Presentation” of Città, Calvino describes the opening and closure of his
book: “il mio libro s’apre e si chiude su immagini di città felici che continuamente
prendono forma e svaniscono, nascoste nelle città infelici (x). This constant appearance
and disappearance in the construction and hiding of forms, not only emulates the twisty
the houses are dismantled; only the strings and their supports remain” (76).
251
“From a mountainside, camping with their household goods, Ersilia's refugees look at the labyrinth of
taut strings and poles that rise in the plain. That is the city of Ersilia still, and they are nothing” (76).
252
“They rebuild Ersilia elsewhere. They weave a similar pattern of strings which they would like to be
more complex and at the same time more regular than the other. Then they abandon it and take themselves
and their houses still farther away” (76).
163
pattern of a spiral, but leads through its meandering routes into that which is concealed.253
Once again from a mathematical structural perspective, the selection of the
number of fifty-five cities is interesting. This is the tenth number in the Fibonacci’s
sequence. 1, 1, 2,3,5,8, 13,21,34,55… This series, originated by Leonardo di Pisa (called
Fibonacci) forms itself by generating each subsequent element with the addition of the
two previous ones.254 Much importance has been given to this series for its approximation
to the structures to many living organisms. One reason behind this may very well be how
it can be used to create a spiral shape, which is so common in nature. Interestingly, the
spiral shape is constantly suggested in Città. In his book Understanding Italo Calvino,
Beno Weiss says: “From several of his writings it would appear that Calvino’s extended
metaphor was the spiral, just as the labyrinth was for Borges” (106).
Also within this Fibonacci series the sum of any ten consecutive numbers equals
eleven times the seventh digit of this sub series, thus granting the reader another potential
link with the Fibonacci series, since eleven is the number of themes, and seven the
number of central chapters, (those that do not contain ten stories, as the enclosing first
and last chapter), but five,255 which is also the number of cities within each of the eleven
categories.
253
Spiral figures suggest the image of that which is continuous and hidden (the two closing themes in
Città).
254
The algorithmic process is as follows: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, 13+21=34,
21+34=55.
255
To illustrate this, the calculations are simple. In the series concerning this book, (1, 1, 2, 3, 5, 8, 13, 21,
34, 55), the seventh number is 13, and 13x11= 143. The sum of this ten number sequence adds up to the
same amount: 1+1+2+3+5+8+13+21+34+55= 143.
164
Another point of mathematical interest regarding the number five acquires
significance within these central seven cities. In each of them five stories portray a
pattern formed by identical countdown sequences, which reduces themselves from five to
one (5 4 3 2 1); each number refers to the presence of each one of the eleven themes (or
categories) by which the book is organized. Thus, as these themes emerge five times,
each one appears for the fifth and last time at the beginning of each of these seven central
chapters, whereas each chapter ends with the entrance of a new theme.
Opening and closure are interlaced: what begins with an ending (closure) ends
with a beginning (opening). This suggests a dual opposite direction, like in a spiral shape,
which can lead towards the center or away from it, or back and forth. Besides, as in a
shell, the center is empty. In Città’s empty and excluded center, the city is suspended
above the void.
Chapter one and chapter nine enclose the other chapters through the combinatorial
game, while reversing each other. Chapter one builds the sequence256 achieved by
combinations on the seven central chapters: 5 4 3 2 1. Whereas chapter nine does the
reverse: it dissolves the sequence257, as in a mirror image or the twisting of a spiral. One
may examine any of these chapters, to see this spiral pattern. In chapter five, to use the
central one as example, the category of “le città sottili” (thin cities) appears for the fifth
256
1
21
321
4321
54321
257
54321
4321
321
21
1
165
and last time (the rubric of “Città sottili 5” pertaining to Octavia), where as the fifth and
last city of this fifth chapter, the city of Melania 258 259has the rubric: “Le città e i morti
1”) introducing the first appearance of the category “Cities and the dead” (perhaps also
pointing to the beginning of the last parts of the book).
As said, opening and closure are interlaced: what begins with an ending (closure)
ends with a beginning (opening).260. In each of these seven central chapters, a theme
appears while another “seems” to disappear in order to give space to a new theme.
Paradoxically, they “appear to disappear”. I intentionally use the word “seems” since the
spiral sequence forms part of an intricate net of interrelationships, combinations and
variations of the themes, which lead to multiple interpretations as they reappear.
Upon observation of the parallelogram, which represents the author’s design of
the book (shown in his letter on the last chapter), one can also perceive that the sequence,
which occurs in the central seven chapters, that is, from the second to the eighth, reveals
a design that is much more than a simple repetition of a serial countdown. The number
five becomes protagonist again in this combinatorial search for form. From chapter two
to chapter eight, one can vertically travel through the same theme: draw a line or move
through the five cities pertaining to that category. On the other hand, horizontally, one
can follow the trail of the five stories (each pertaining to a theme) in each chapter. That
258
In Melania, as already discussed, “every time you enter the square, you find yourself caught in a
dialogue…one role may be doubled, multiplied, assigned to a hundred, a thousand inhabitants of Melania:
three thousand for the hypocrite, thirty thousand for the sponger. a hundred thousand kings sons fallen in
low estate and awaiting recognition” (80-81).
259
“If you look into the square in successive moments, you hear how from act to act the dialogue changes,
even if the lives of Melania”s inhabitants are too short for them to realize it” (81).
260
The reversal of last and first chapters is analyzed further in this section.
166
is, in this combinatorial map, we may vertically trace the themes, and horizontally, travel
through the chapters.
Another intriguing aspect of this, at least for some, apparent symmetry, is that it
leads to a particular central subject. Just as in a seashell, the center of this book reveals
and hides an empty space, a void. The city of Bauci is invisible: its absence marks its
unique quality.
In his presentation to Città Calvino addresses this interpretation given by the
critics: “Hanno detto che è nel centro che bisogna cercare: e hanno trovato un’immagine
di assenza, la città chiamata Bauci” (Città X).
The traveller, according to Marco’s storytelling, arrives at Bauci after travelling
for seven days. But the city cannot be seen. High above the clouds, only the flamingo
legs, which support it, are visible.
Dopo aver marciato sette giorni attraverso boscaglie, chi va a Bauci non
riesce a vederla ed è arrivato. I sottili trampoli che s'alzano dal suolo a
gran distanza l'uno dall'altro e si perdono sopra le nubi sostengono la città.
Ci si sale con scalette. A terra gli abitanti si mostrano di rado: hanno già
tutto l'occorrente lassù e preferiscono non scendere. Nulla della città tocca
il suolo tranne quelle lunghe gambe da fenicottero a cui si appoggia e,
nelle giornate luminose, un'ombra traforata e angolosa che si disegna sul
fogliame (77).261
The fantastic description, once again acquires a humorous turn through those details that
approximate reality: “On the ground the inhabitants rarely show themselves: having
already everything they need up there, they prefer not to come down” (77).
261
“After a seven days' march through woodland, the traveler directed toward Baucis cannot see the city
and yet he has arrived. The slender stilts that rise from the ground at a great distance from one another and
are lost above the clouds support the city. You climb them with ladders. On the ground the inhabitants
rarely show themselves: having already everything they need up there, they. Nothing of the city touches the
earth except those long flamingo legs on which it rests and, when the days are sunny, a pierced, angular
shadow that falls on the foliage” (77).
167
The city exists, even if it cannot be seen; but it is located elsewhere, above. This
leads to conjecture. The possible explanation for Bauci’s invisibility is triple, but points
to the third option: the inhabitants, from the distance, contemplate the very absence of the
city, “fascinated” .“Tre ipotesi si dànno sugli abitanti di Bauci: che odino la Terra; che la
rispettino al punto d'evitare ogni contatto; che la amino com'era prima di loro e con
cannocchiali e telescopi puntati in giù non si stanchino di passarla in rassegna, foglia a
foglia, sasso a sasso, formica per formica, contemplando affascinati la propria assenza”
(77).262
Similarly, the city of Irene is seen in a distance and its inhabitants also feel no
need to visit it: “Not that they have any intention of going there (in any case the roads
winding down to the valley are bad), but Irene is a magnet for the eyes and thoughts of
those who stay up above” (124). Yet, they look at it from a different viewpoint. Irene is
seen from atop: “Travelers on the plateau, shepherds shifting their bocks, bird-catchers
watching their nets, hermits gathering greens: all look down and speak of Irene.”263
Irene is not only seen from above but also, only from distant perspective. Distance
is required for perspective to be able to detect patterns. “The form of things can be
discerned better at a distance” (Città 98). (“Le forme delle cose si distingue meglio in
262
“There are three hypotheses about the inhabitants of Baucis: that they hate the earth; that they respect it
so much they avoid all contact; that they love it as it was before they existed and with spyglasses and
telescopes aimed downward they never tire of examining it, leaf by leaf, stone by stone, ant by ant,
contemplating with fascination their own absence” (77).
263
“Quelli che guardano di lassù fanno congetture su quanto sta accadendo nella città, si domandano se
sarebbe bello o brutto trovarsi a Irene quella sera. Non che abbiano intenzione d’andarci – e comunque le
strade che calano a valle sono cattive – ma Irene calamita sguardi e pensieri di chi sta là in alto” (124).
168
lontananza” 99).264 “At this point Kublai Khan expects Marco to speak of Irene as it is
seen from within. But Marco cannot do this… For that matter, it is of slight importance:
if you saw it, standing in its midst, it would be a different city; Irene is a name for a city
in the distance, and if you approach, it changes” (124).265
In an article titled “Il mihrab” ”(Collezioni di sabbia 219-222), Calvino describes
how, upon contemplating admirable designs, rich with geometric figures, leading to a
central cavity, like in a shell, he concludes that everything must lead to an object. But this
object is not there. In fact, he explains, this is its most important characteristic: not being.
Dopo essere rimasto un bel pezzo a contemplare il mihrab, mi sento in
dovere di giungere a una qualche conclusione. Che potrebbe essere questa:
l’idea di perfezione che l’arte insegue, la sapienza accumulata nella
scrittura, il sogno di appagamento d’ogni desiderio che si esprime nello
sfarzo degli ornamenti, tutto rimanda a un solo significato, celebra un solo
principio e fondamento, implica un solo ultimo oggetto. Ed è un oggetto
che non c’è. La sua sola qualità è quella di non esserci (220).
According to the author, what he learned from his travels to distant lands and
times, was that which is most important: the empty spaces: “Questo avevo creduto di
capire in quel mio lontano viaggio a Ispahan; che la cosa più importante al mondo sono
gli spazi vuoti” (Sabbia 221). Calvino relates the void, the empty space with fantasy and
game “Il vuoto ha le sue fantasie i suoi giochi” (Sabbia 221). This conception is in
agreement with Kublai Khan’s appreciation of the empty spaces between cities in Città,
which Marco Polo addresses later on his narrative game, by telling yet another fantastic
264
In Hermit in Paris says a similar idea concerning Paris: “Maybe to write about Paris I ought to leave, to
distance myself from it, if it is true that all writing starts from a lack or an absence…” (167).
265 “A questo punto Kublai Kan s’aspetta che Marco parli d’Irene com’è vista da dentro. E Marco non può
farlo: quale sia la città che quelli dell’altipiano chiamano Irene non è riuscito a saperlo; d’altronde poco
importa: a vederla standoci in mezzo sarebbe un’altra città; Irene è un nome di città da lontano, e se ci si
avvicina cambia “ (124). 169
story, that of Cecilia. “Tu mi rimproveri perché ogni mio racconto ti trasporta nel bel
mezzo d’una città senza dirti dello spazio che s’estende tra una città e l’altra: se lo
coprano mari, campi di segale, foreste di larici, paludi. Ti risponderò con un racconto. Per
le vie di Cecilia…” (152).266 Besides, this idea is more than implicit in Calvino’s
definition of literature as a combinatorial game, as a search for space that will allow the
imagination to keep on playing the game, creating, designing new forms and letting
fantasy breath and leading to continuity.
Also on “Il mihrab”, Calvino describes, in contrast, the qualities of a hypothetical
city, which could be, in part, those of the ideal city for which Polo’s stories, in Città, aim:
“Forse una città che è stata fatta seguendo una felice disposizione dei pieni e dei vuoti si
presta a essere vissuta con felice disposizione di spirito” (Sabbia 222). The word
“disposizione” is subtly employed with a double meaning to relate the optimal
organization of the city spaces (both empty and filled) and the good disposition the
inhabitants would experience in such a city.
In “Le effimere nella fortezza” (Sabbia 83-85), the void can also turn out to
acquire “heaviness” as in the “vuoto-pieno”, for which the only solution is the lightness,
rapidity of that which is “sottile”, as in Città: “il vuoto-pieno che può essere risolto solo
da ciò che è leggero e rapido e sottile” (Sabbia 83).267
266
“You reproach me because each of my stories takes you right into the heart of a city without telling you
of the space that stretches between one city and the other, whether it is covered by seas, or fields of rye,
larch forests, swamps. 1 will answer you with a story. In the streets of Cecilia…” (152).
267
Moreover, in “I mille giardini”, Calvino describes the universe as “un equilibrio di pieni e di vuoti”
(Sabbia 190).
170
In this article’s description, the geometric steel figures, which seem to declare
war, do not constitute that which defends the space: “Sui bastioni verdi dal Belvedere una
palizzata di forme geometriche d’acciaio brunito irta di lance a punta levata o confitta al
suolo può evocare una guerra barbarica o extraterrestre; ma subito ci rendiamo conto che
è piazzata a difendere uno spazio in cui la forza che vince è quella interiore, l’ostinazione
è fatta da linee sottili, la sfida è sostenuta dall’ironia” (Sabbia 84).
As described in the filiformi cities, subtle, delicate lines uphold the interior space
of the mind: it is irony that sustains, withstands the challenge. In “The situation in 1978”,
Calvino is asked what roles irony plays for him. To which he author answers: “ Irony
warns that what I write must be read with a distracted air, a mood of considerable
lightness…the things that count are particularly those I say with irony” (Hermit in Paris
189). But it is not just lightness that is involved, as he, underlining his usual twists and
turnings around, adds: “Irony always warns of the other side of the coin.” (190). Duality
and reversals continue to play their mathematical games.
The predominance of the spiral shape can be seen in the design of the book and of
the cities within the book. It designs a pattern, a structure in motion, constantly changing,
evolving into new forms, creating lightness. For the ancient Egyptians the spiral denoted
“cosmic forms in motion, or the relationship between unity and multiplicity”.268 In
traditional interpretation the “creative” spiral unwinding around a point while moving
further away from it symbolizes growth and represents the evolution of evolution and the
unfolding of creation. In this symbolic expansion and regression, the spiral “circles from
one stage of development to another”. Moving in a spiral direction through space and
268
Weiss cites J.E. Cirlot Dictionary of Symbols .
171
time, the spiral shows “the relationship between his own unity… and that of multiple
other forms belonging to the cosmos and to other species” (Weiss 107).
The city of Isidora (“Cities and memory 2”) portrays a traveler whom upon
desiring a city, arrives at one designed by spirals within spirals. (In turn, the first city
within the category of “Cities and Desire”, Dorotea, follows Isidora.) “All’uomo che
cavalca lungamente per terreni selvatici viene desiderio d’una città. Finalmente giunge a
Isidora, città dove i palazzi hanno scale a chiocciola incrostate di chiocciole
marine…”(8). (“When a man rides a long time through wild regions he feels the desire
for a city. Finally he comes to Isidora, a city where the buildings have spiral staircases
encrusted with spiral seashell…” 8). In these twisty structures, the desire of a future city
seems to coincide, concur with the present and coexist with the past.
Also Fedora, “the gray stone metropolis”, as mentioned, has at its center “a metal
building with a crystal globe in every room”. The globes, different models of Fedora,
“are the forms the city could have taken if, for one reason or another, it had not become
what we see today” (32). These forms become more complex as the visitors envision
their reflection on the water, fancy a perspective from the heights, and even imagine their
enjoyment while playfully gliding down a spiral:
The building with the globes is now Fedora's museum: every inhabitant
visits it, chooses the city that corresponds to his desires, contemplates it,
imagining his reflection in the medusa pond that would have collected the
waters of the canal (if it had not been dried up), the view from the high
canopied box along the avenue reserved for elephants (now banished from
the city), the fun of sliding down the spiral, twisting minaret (which never
found a pedestal from which to rise) (32).269
269
“Fedora ha adesso nel palazzo delle sfere il suo museo: ogni abitante lo visita, sceglie la città che
172
Even within the category of “Cities and the Sky”, one encounters cosmic growth
and alterations, evolving into winding, spiral forms: “The astronomers, after each change
takes place in Andria, peer into their telescopes and report a nova's explosion, or a remote
point in the firmament's change of color from orange to yellow, the expansion of a
nebula, the bending of a spiral of the Milky Way” (151).270
Calvino knew the visual image of the Milky Way as a spiral as well as Dante’s spiral
downward journey to Hell and his opposite ascending spiral one to Purgatory (Weiss
106-107).
In regards to these intergalactic forms, Calvino’s book, Cosmicomiche, includes a
particular story “La spirale” (“The Spiral”), entirely devoted to the creative life of a
mollusk in its shell “all twisted into a spiral.” The spiral represents growth and creativity.
Also, in “Stendhal’s Knowledge of the “Milky Way” (Uses of literature 266-283),
Calvino refers to Stendhal’s method as “based on individual experience in all its
unrepeatable uniqueness” (266), his ‘mind very opposed to a systematic order” (268) and
intriguingly, his “mathematics immediately become very complicated” (269).
Simply because existence is dominated by entropy, by dissolution into
instants and impulses like corpuscles without form of their own links with
others, he thinks that the individual realizes himself according to a
principle of conservation of energy, or, rather, of the continual
reproduction of changes of energy. This is near to understanding that
corrisponde ai suoi desideri, la contempla immaginando di specchiarsi nella peschiera delle meduse che
doveva raccogliere le acque del canale (se non fosse stato prosciugato), di per-correre dall’alto del
baldacchino il viale riservato agli elefanti (ora banditi dalla città), di scivolare lungo la spirale del minareto
a chiocciola (che non trovò più la base su cui sorgere” (32).
270
“Gli astronomi scrutano coi telescopi dopo ogni mutamento che ha luogo in Andria, e segnalano
l’esplosione d’una nova, o il passare dall’arancione al giallo d’un remoto punto del firmamento,
l’espandersi di una nebula, il curvarsi d’una spira della via lattea” (151).
173
entropy will win out in the end, and that of the universe with all its
galaxies nothing will be left but a whirlwind of atoms in space” (Uses
283).
In Città, the Khan, towards the end of the book, also uses the image of the spiral.
At this final moment, he is found “leafing through the atlas”, looking at maps of infernal
cities, particularly the amorphous, continuous ones. This appears right before Polo’s final
statement in the book, regarding the inferno: “Tutto è inutile se l’ultimo approdo non può
essere che la città infernale, ed è là in fondo che in una spirale sempre più stretta, si
richiude la corrente” (164). (“It is all useless, if the last landing place can only be the
infernal city, and it is there that, in ever –narrowing circles, the current is drawing
us”165). Yet, on another occasion concerning the spiral shape, Calvino’s article on
Francis Ponge refers to the sea shell as “the last form of happiness“: “che sia la lumaca
l’ultima forma di felicità possibile” (Saggi 1407).
This shape projects both the concept of continuity and that of the hidden, the last
two themes in Città. The last city-story of Le città invisibili describes Berenice, a
“Hidden City” (also the last of the series of eleven categories present in the book).
(Regardless of how “unjust” Berenice might seem, Polo insists that he must tell of the
“other” Berenice, hidden, clandestine, its just fair version:
I should not tell you of Berenice, the unjust city… Instead, I should tell
you of the hidden Berenice, the city of the just, handling makeshift
materials in the shadowy rooms behind the shops and beneath the stairs,
linking a network of wires and pipes and pulleys and pistons and
counterweights that infiltrates like a climbing plant among the great
cogged wheels (when they jam, a subdued ticking gives warning that a
new precision mechanism is governing the city) (161).271
271
“Anziché dirti di Berenice, città ingiusta, …dovrei parlarti della Berenice nascosta, la città dei giusti,
armeggianti con materiali di fortuna nell’ombra di retrobotteghe e sotto scale, allacciando una rete di fili e
tubi e carrucole e stantuffi e contrappesi che s’infiltra come una pianta rampicante tra le grandi ruote
dentate (quando queste s’incepperanno, un ticchettio sommesso avvertirà che un nuovo esatto meccanismo
174
Polo’s persistence is emphasized through repetition: he continually uses opposition to
create contrast and to highlight, to underline that, which is essentially noteworthy.
(Recurrently, he continues to say that, instead of telling - or not telling- something, he
should tell you something else: “I should not tell you”, “Instead I should tell you”, or
rather, “anziché”… “dovrei dirti”.)
Anziché rappresentarti le vasche profumate delle terme sdraiati sul cui
bordo gli ingiusti di Berenice intessono con rotonda eloquenza i loro
intrighi e osservano con occhio proprietario le rotonde carni delle
odalische che si bagnano, dovrei dirti di come i giusti, sempre guardinghi
per sottrarsi alle spiate dei sicofanti e alle retate dei giannizzeri, si
riconoscano dal modo di parlare, specialmente dalla pronuncia delle
virgole e delle parentesi; dai costumi che serbano austeri e innocenti
eludendo gli stati d’animo complicati e ombrosi; dalla cucina sobria ma
saporita, che rievoca un’antica età dell’oro: minestrone di riso e sedano,
fave bollite, fiori di zucchino fritti (161).272
At a certain point, Polo’s description implies what could be the process employed in
mathematical analysis: “From these data it is possible to deduce an image of the future
Berenice, which will bring you closer to knowing the truth than any other information
about the city as it is seen today” (162). Still, the intention, in contrast, is to point out that
something else, that otherness, that which is hidden within. “You must nevertheless bear
in mind what I am about to say to you: in the seed of the city of the just, a malignant seed
is hidden, in its turn […]” (162). 273
governa la città)” (161).
272
“Instead of describing to you the perfumed pools of the baths where the unjust of Berenice recline and
weave their intrigues with rotund eloquence and observe with a proprietary eye the rotund mesh of the
bathing odalisques, I should say to you how the just, always cautious to evade the spying sycophants and
the Janizaries' mass arrests, recognize one another by their way of speaking, especially their pronunciation
of commas and parentheses; from their habits which remain austere and innocent, avoiding complicated
and nervous moods… “(161).
273
“Da questi dati è possibile dedurre un’immagine di Berenice futura, che ti avvicinerà alla conoscenza
del vero più d’ogni notizia sulla città quale oggi si mostra. Sempre che tu tenga conto di ciò che sto per
175
Within the unhappiness of the (continuous) cities, the “hidden cities” alternate
images of happy cities. Marozia “consists of two cities, the rat's and the swallow's; both
change with time, but their relationship does not change; the second is the one about to
free itself from the first” (155).274 Similar recursive structures appear throughout the
hidden cities. Olinda, which grows concentrically, has the unique quality of growing by
keeping its wall expanded; yet proportionally creates space for the other reduced Olindas
to come:
The old walls expand bearing the old quarters with them, enlarged, but
maintaining their proportions on a broader horizon at the edges of the city;
they surround the slightly newer quarters, which also grew up on the
margins and became thinner to make room for still more recent ones
pressing from inside; and so, on and on, to the heart of the city, a totally
new Olinda which, in its reduced dimensions retains the features and the
flow of lymph of the first Olinda and of all the Olindas that have
blossomed one from the other; and within this innermost circle there are
already blossoming-though it is hard to discern them-the next Olinda and
those that will grow after it (129-130).275
Marco appeals for attentiveness: “Un’altra città ingiusta, pur sempre diversa dalla
prima, sta dunque scavando il suo spazio dentro il doppio involucro delle Berenici
ingiuste e giuste”.
dirti: nel seme della città dei giusti sta nascosta a sua volta una semenza maligna; la certezza e l’orgoglio
d’essere nel giusto – e d’esserlo più di tanti altri che si dicono giusti più del giusto – fermentano in rancori
rivalità ripicchi, e il naturale desiderio di rivalsa sugli ingiusti si tinge della smania d’essere al loro posto a
far lo stesso di loro” (162).
274
“L’oracolo sbagliava? Non è detto. Io lo interpreto in questo modo: Marozia consiste di due città: quella
del topo e quella della rondine; entrambe cambiano nel tempo; ma non cambia il loro rapporto: la seconda è
quella che sta per sprigionarsi dalla prima” (155).
275
“Mantenendo le proporzioni su un più largo orizzonte ai confini della città; essi circondano i quartieri
un po’ meno vecchi, pure cresciuti di perimetro e assottigliati per far posto a quelli più recenti che premono
da dentro; e così via fino al cuore della città: un’Olinda tutta nuova che nelle sue dimensioni ridotte
conserva i tratti e il flusso di linfa della prima Olinda e di tutte le Olinde che sono spuntate una dall’altra; e
dentro a questo cerchio più interno già spuntano – ma è difficile distinguerle – l’Olinda ventura e quelle che
cresceranno in seguito” (130).
176
Having said this, I do not wish your eyes to catch a distorted image, so I
must draw your attention to an intrinsic quality of this unjust city
germinating secretly inside the secret just city: and this is the possible
awakening-as if in an excited opening of windows of a later love for
justice, not yet subjected to rules, capable of reassembling a city still more
just than it was before it became the vessel of injustice. But if you peer
deeper into this new germ of justice you can discern a tiny spot that is
spreading like the mounting tendency to impose what is just through what
is unjust and perhaps this is the germ of an immense metropolis… (162163).276
Finally, Marco concludes with a logical yet startling explanation. “Dal mio discorso avrai
tratto la conclusione che la vera Berenice è una successione nel tempo di città diverse,
alternativamente giuste e ingiuste. Ma la cosa di cui volevo avvertirti è un’altra: che tutte
le Berenici future sono già presenti in questo istante, avvolte l’una dentro l’altra, strette
pigiate indistricabili” (163).277
This last city becomes a prelude to the end of the book: the inferno of the living,
which is like the future Berenices, already present.
The same image of the spiral may acquire different, even opposite meanings. This
is due to a structure that allows both centripetal and centrifugal motion. “What matters is
not the enclosure of the work within a harmonious figure, but the centrifugal force
produced by it - a plurality of language as a guarantee of a truth that is not merely partial”
276
“Detto questo, se non voglio che il tuo sguardo colga un’immagine deformata, devo attrarre la tua
attenzione su una qualità intrinseca di questa città ingiusta che germoglia in segreto nella segreta città
giusta: ed è il possibile risveglio – come un concitato aprirsi di finestre – d’un latente amore per il giusto,
non ancora sottoposto a regole, capace di ricomporre una città più giusta ancora di quanto non fosse prima
di diventare recipiente dell’ingiustizia. Ma se si scruta ancora nell’interno di questo nuovo germe del giusto
vi si scopre una macchiolina che si dilata come la crescente inclinazione a imporre ciò che è giusto
attraverso ciò che è ingiusto, e forse è il germe d’un’immensa metropoli …” (162-163).
277
“From my words you will have reached the conclusion that the real Berenice is a temporal succession of
different cities, alternately just and unjust. But what I wanted to warn you about is something else: all the
future Berenices are already present in this instant, wrapped one within the other, confined, crammed,
inextricable” (163).
177
(Six Memos 117).278
Actually, in Physics the only force acting upon a rotating object is always a pull
towards the central axis, i.e. centripetal, whereas the centrifugal one, pushing away, even
though perceived within the rotating object, is considered as non-existent, imaginary.
However, an observation could be made here, in regards to this study: all forces, even
gravity, measurable as they may be, are, nevertheless, invisible.
Mathematics is, once more, used as a means for approaching and acknowledging
what is being perceived. Multiplicity of fragments within the combinatorial game leads to
continuity (through or (through overlapping, reversals, lack of limits…). These
complexities coexist. The continuous cities lead to hidden cities. The figure of the spiral
becomes a mathematical approach to visualize this as a tendency or pattern, instead of
perceiving it as problem without solution.
As mentioned earlier, Calvino revealed his writing process to be “more often than
not involved the subtraction of weight…Maybe I was only then becoming aware of the
weight, the inertia, the opacity of the world - qualities that stick to the writing from the
start, unless one finds some way of evading them” (Six Memos 3-4).279 This awareness
triggers a search, an exploration of different techniques. In mathematical concepts
Calvino’s creative mind finds tactics with which to elude heaviness and impenetrability.
Calvino’s cities point to the reader this process of “becoming aware”; through the
combinatorial game and mapping he finds a way, method, approach, technique to fight, to
278
“Anche se il disegno generale è stato minuziosamente progettato, ciò che conta non è il suo chiudersi in
una figura armoniosa, ma è la forza centrifuga che da esso si sprigiona, la pluralità dei linguaggi come
garanzia d'una verità non parziale” (Lezioni 127).
279
“Forse stavo scoprendo solo allora la pesantezza, l'inerzia, l'opacità del mondo: qualità che s'attaccano
subito alla scrittura, se non si trova il modo di sfuggirle” (Lezioni 8).
178
challenge, to create forms that evade, elude “the weight, the inertia, the opacity”. Such
forms are embodied through the network, the spiral, the labyrinthine, evasive shapes
inspired on lightness. There is also the image of verticality, of cities, which ascend
defying gravity and representing weightlessness and lightness, opposing density and
opacity. 280
As previously discussed, chapter one and chapter nine “enclose” the other
chapters through the combinatorial game. Chapter one builds the sequence achieved:
54321, by combinations. Whereas chapter nine does the reverse: it dissolves the
sequence, as in a spiral, which can grow one way and unravel in the opposite direction. It
should be noted how the enclosing chapters, one and nine, form a triangle with the ten
stories they contain. Yet these triangles are non-symmetrical reversals to one another, as
in a mirror image. They continuously reflect one another so that a beginning and an
ending, opening and closure constantly occur. ”Il viaggio potrebbe compiersi ugualmente
attraverso uno specchio cui non resta altro da specchiare che la propria cornice oppure
attraverso una finestra che s’affacci sul fuori d’ambe due le parti” (“I segni alti (per
Fausto Melotti)”, Saggi II: 1971).
Through the mirror image the possibility of otherness, of the elsewhere, of the
other side, is open, at least in our imagination. (In “Il rovescio del sublime”, Calvino
describes the gardens of Kyoto as “prospettive in salita o in dicessa suggeriscono spazi
che non ci sono…comunicano l’idea di vivere a parte da quello che è il mondo, al riparo
280
Falls are vertical motions like the movement of the lighter cities. Verticality is related to a movement
into “otherness”, as Alice in Wonderland; an elsewhere which is already present as a mirror projection.
179
dalla storia catastrofica e incongrua” (Sabbia 177-179).281
The book has a beginning and an ending. But due to its duplicity, as well as its
reversibility of space, its twisty patterned webs, an openness to continue searching and
creating spaces is suggested even by Marco Polo’s (and the book’s) final words: “cercare
e saper riconoscere chi e cosa, in mezzo all’inferno, non è inferno, e farlo durare, e dargli
spazio.”282 The book ends by his giving us a double option; yet Marco warns us to
“aguzzare lo sguardo.”
In Città all the stories are told in Kublai’s garden (the garden being like another
story space). Just as in the gardens of Kyoto (on the essay mentioned above) “Sta a noi
vedere questo giardino come lo spazio d’un altra storia…“ (180).
Again, it seems that what is important is to discern and understand that which
may not be obvious, stagnant, imposing its weight and inertia; but to question and explore
that which may be hidden, which escapes the system, that “otherness” which is present in
its lightest forms. However evasive, the pursuit continues.
Nella mia esperienza la mia spinta a scrivere è sempre legata a la
mancanza di qualcosa…qualcosa che si sfugge. E siccome conosco bene
questo tipo di spinta, mi sembra di poterlo riconoscere anche nei grandi
scrittori le cui voci sembrano giungere dalla cima d’una esperienza
assoluta . Quello che essi trasmettono è il senso dell’approccio
all’esperienza, più che il senso delle esperienze raggiunte; il loro segreto è
sapere conservare intatta la forza del desiderio”(Saggi II 1874).
281
“The city is the place where alterity discloses itself” …”The place of the revelation of alterity”. Modena
(120) cites Celati as quoted by Barenghi and Belpolity (Ali Babà: Progetto di una rivista, 1968-1972).
.282Also in “Il rovescio del sublime”: “Crearsi uno spazio ed un tempo per riflettere e immaginare e studiare
presuppone un’accumulazione….Ogni progetto o immagine che permetta di tendere a un altro modo
d’essere fuori dell’ingiustizia che circonda porta il marchio della ingiustizia senza la quale non sarebbe
stato concepito” (Sabbia 180).
180
Evasiveness is also explored by mathematics. According to the aforementioned
Gödel’s Incompleteness Theorem, for any system, something emerges which is not part
of the system, or at least, cannot ne explained by itself. Consequently, that which escapes
the system becomes essential. (This was implicit, the hypothesis, fundamental for this
research on Città.) Calvino’s cities comprise, give form to vivid and rich examples of this
theorem.
In the very last lines of Dante’s Divine Comedy (“Paradiso” Canto XXXIII), an
analogy is drawn between the pursuits of the geometer (mathematics) and the poet
(literature):
Qual è ’l geomètra che tutto s’affige
per misurar lo cerchio, e non ritrova
pensando, quel principio ond’ elli indige,
tal era io a quella vista nova:
veder voleva come si convenne
l’imago al cerchio e come vi s’indova;
ma non eran da ciò le proprie penne:
se non che la mia mente fu percossa
da un fulgore in che sua voglia venne.
A l’alta fantasia qui mancò possa;
ma già volgeva il mio disio e ’l velle,
sì come rota ch’igualmente è mossa,
181
l’amor che move il sole e l’altre stelle.283
The correlation between the geometer and the poet is based on their pursuit for that which
cannot be expressed (inexpressible) or said (ineffable). Similarly, a constant search is
ever present in Calvino’s writing. He describes the process of both writing and reading
as a quest, of literature trying to say, that which is not in the dictionary.
Calvino’s interdisciplinary approach opens new possibilities to his writing and its
interpretations. In his intellectual and artistic pursuit he manages to combine literature
and mathematics, or manages to convey images of mathematics integrated into his
writing to the point that critics have pointed out how “the pattern becomes the story”284
and that in his writing “pattern is paramount”285.
In Città this search for form, the “filigrana così sottile” provided by Marco’s
stories could be seen to acquire, among other patterns or forms, that of the spiral,
constructed of minuscule fragments like the shell, yet simultaneously conveying
continuity and perhaps most important, the suggestion of something hidden. What could
have been a labyrinth becomes for Calvino a structure which projects lightness,
creativity, growth, bidirectional motion and a way out. Most importantly, it inspires the
desire to keep searching, to listen attentively within forms concealing inside whispers of
283
“As the geometrician, who endeavors To square the circle, and discovers not, By taking thought, the
principle he wants Even such was I at that new apparition, I wished to see how the image to the circle
Conformed itself, and how it there finds place But my own wings were not enough for this Had it not been
that then my mind there smote A flash of lightning, wherein came its wish. Here vigor failed the lofty
fantasy: But now was turning my desire and will, Even as a wheel that equally is moved The Love which
moves the sun and the other stars.”
284
Ricci 144.
285
Carter 111.
182
an ocean of stories, stories that appear at a distant, but are already present, and above all
live somewhere hidden potentially within our minds.
In order to create these mental visualizations, mathematics is involved in the
construction of these city forms. Mathematics becomes a source for structures, strategies,
methods to deal with and try to convey the concepts of lightness, visibility, exactitude,
multiplicity and even evasiveness.
From his concept of narrative as a combinatorial game, Calvino accomplishes
more than creating a multiplicity of stories with a few elements. Not only does he manage
to carefully design a filigree of interconnected nets enticing his reader to find an exit. But
at every step, he emphasizes that which the combinatorial system triggers, that which
escapes the system. Moreover, he integrates this element in such a way as to make it the
essence of each city story. Again, Gödel’s Incompleteness Theorem states that in any
system there is something that is not part of the system. Paradoxically, this is how
Calvino makes a system consistent: the unexpected becomes part of the whole.286 Both
the writer and the mathematician successfully integrate these complex ideas, by
generating the necessary leap, which requires agility of mind, perspective, visibility,
exactitude, rapidity and lightness.
Calvino’s fantastic stories –webs, spirals, carpets- move, constantly changing
forms in our imagination, swiftly winding and unwinding city stories, crammed or light,
so that we may carry this book-city, within ourselves, to open or close, as the journey
286
According to Gödel, if the system is consistent, it cannot be complete. This consistency cannot be
proven within the system. There is something that will be not be provable, or cannot be demonstrated.
Peculiarly, Calvino’s last or sixth memo, which was never finished due to his death, was to be on
consistency.
183
continues, tracing paths that intersect, opening our minds to other striking possibilities,
apparently dissimilar, invisible, yet present in the realm of literature and mathematics. “E
nella tessitura dei tappeti che i nomadi depositano la loro sapienza: oggetti variegati e
leggeri che si stendono sul nudo suolo dovunque ci si ferma a passare la notte e si
arrotolano al mattino per portarli via con sé insieme a tutti i propri averi sulla gobba dei
cammelli” (Sabbia 233).287
287 “It
is in the weaving of the carpet that nomads deposit their knowledge: variegated and light objects
which they extend over the bare floor wherever they stop to spend the night and roll-up in the morning to
take them with themselves together with all their belongings over the camel’s gibe”.287
184
Conclusion
“Mathematics compares the most diverse phenomena and discovers the
secret analogies that unite them” (Joseph Fourier 1768 -1830).
This study was intended as an interdisciplinary approach in the analysis of the
interrelationship between literature and mathematics. Among Italo Calvino’s vast
quantity of writings, Le città invisibili proved to be a fertile ground full with implications
to be explored from this perspective. The findings have not only corroborated the original
hypothesis that mathematical concepts, defined as a science of structures, are present in
the creative construction of Città, but also lead to potential repercussions as they share
with art and literature the aesthetic pursuit of form. Math concepts were uncovered
permeating in a light-footed manner filtering within the book’s fantastic stories.
Beyond showing the presence of mathematical concepts, this study draws a
connection between Calvino’s conception of narrative as a combinatorial art and the
author’s literary values expressed in his Six Memos for the Next Millennium. Calvino’s
definition of literature as a combinatory game was a starting point. This point of
departure became a bridge leading to other essential links. Through a mathematical
approach to Le città invisibili, this study unveils and correlates the values the author
attributes to literature in his last legacy, Six Memos for the Next Millennium. The
selection of this collection of essays, among others proved to be appropriate. It served as
a guide, a lead that kept on manifesting itself through the text, as if Calvino’s ideas of Six
Memos were already germinating in Città.
This last, unfinished collection of essays reflects Calvino’s evolved concepts in
many respects, but my intention here was to focus on the mathematical aspect. An aspect
essentially interrelated to others, such as science, architecture, but that stands on its own,
185
particularly for being the base, the foundation to other approaches. The endeavor to
extract these basic concepts and analyze them proved to be both meaningful and valuable.
The study begins by analyzing the overall structure as designed by the author and
delineated in the index. The book has a fragmentary nature. This was viewed and
analyzed as a modular strategy fundamental to a combinatorial process. Calvino’s
preference for short modular pieces is concurrent with the concepts of quickness
(rapidity) and exactitude as exposed in Six Memos. Mathematically the interconnections
are quite precise. The succinct, brief petite poèmes en prose, so carefully measured,
rhythmically calculated, can ultimately be perceived as “small mathematical poems”.
According to the author, exactitude bifurcates (Six Memos) and, as explored in
this study, this binary branching leads to multiplicity. Based on duplicity and made
explicit in Città where everything as the author reveals is a double, upon examination this
binary structure is the key to further mathematical relationships. Multiplicity is a concept
developed not just in the fanning out of possible routes as suggested by the index
combinatorial game, but further developed in the cities as examples of variety. Yet, this
multiplicity must be contained. This is one of the functions of the frame: to unify
multiplicity. Besides embedding the various city stories, the frame creates a selfreflective structure which comments the process of binary oppositions expressed in the
dialogues between the Emperor and Marco Polo.
Mapping, a main mathematical strategy becomes evident in the book from its
combinatorial index. This map functions as a blueprint to the other main mathematical
aspect such as motion. Mobility, as change of space in time, reveals itself through this
mapping. The structure delineated grants the reader the freedom, to “move” around the
186
book space by imagining, tracing other paths in between or within the city spaces. As we
traveled through the cities, the network of possibilities became more observable.
Mapping allowed not just the overall structure to be “mobile” but due to the “oulipian”
nature, the binary system of the text gives way to multiplicity and visibility.
On the second chapter, the geometric concepts beginning with the basic point to
the binary relation of the line, which then multiplies into more complex figures, serve to
visualize the cities. Stories evolve into fantastic forms. Through a process of connections
and contrasts these figures or shapes in space become more complex due to the nature of
their relationships.
Duplicity, as a mathematical scheme gives way to reversals, inversions; by
doubling itself, it leads to multiplicity and exponential growth; by going in the opposite
direction, to division and integration. All of these constitute techniques used by
mathematicians to create new forms and structures. However, the opposite is also true.
From a mathematical perspective, the cities could be said to draw abstract mathematical
concepts of numbers, duplicity, and multiplicity as visual images in our minds. Numbers
count in unpredictable ways. This study points out how in Città mathematical concepts
such as numbers are part of the story (rhythmically, visually and thematically).
Another fundamentally binary correlation, which proved to be crucial to the book,
is that established by a model. In the case of Città, the selection of Venice as a non
Euclidean model was essential to accommodate the fluidity of the system. As we trace
relationships, we become involved in the web, the network established by the
multiplicity. Hence, as we travel - read through the book - complexities evolve. At times,
limits appear to be unclear and puzzling due to lack of distinction. Discrete elements
187
merge into a continuum. The transition from discrete notions to the more complex
conceptions of overlapping boundaries, infinity and continuity, proved to be yet another
method where underlying mathematical ideas were effective constituents for the
construction of the narrative.
Moving in space, involves game playing in a non-static ever changing space
which expands spontaneously challenging the rules it has established and by which it has
been established. What escapes the system is paradoxically what keeps its essence. The
only thing that remains constant is change.
Among these spaces in motion, the image of the spiral provides an approximation
to a form, by which we can mentally visualize one possible perspective and interpretation
of the book space. Aesthetically, its resemblance to a Fibonacci series, gives illusion of
openness and enclosure. It includes both the fragmentary and the continuous, twisting and
turning, to accommodate for the ironic and contrasting viewpoints that run through the
pages.
Finally, these paths, parallel or interlaced lead to lightness. Particularly, the
center, which paradoxically is absent as in a shell, portrays weightless images through
such mathematical concepts as verticality, suspension, and networks. Lightness as
described by the author is found mostly in these “filiforme”, threadlike figures, which to
him are the most “luminous”. The visual aspect of lighter architectural constructs is not
just “strangely linked to the city” as Calvino says in Città’s Presentation, but also,
according to the findings and implications of this study, curiously interconnected to
mathematics. In this subtraction of weight – in this lightness - it is also the aesthetic
aspect which mathematics and literature share.
188
In this study’s findings we have seen that the values attributed to literature by
Calvino in Six Memos, namely, exactitude, visibility, quickness, lightness, multiplicity
and even coherence are also recognized in mathematics. Conceivably, the more explicit
mathematical correlations were uncovered within the themes of exactitude, quickness and
multiplicity. Particularly captivating are the associations implied by connections between
visibility, lightness and mathematics. Still, within all these topics, mathematical
reasoning (even its paradoxes) revealed unsuspected analogues with literary creation.
The implications from the results of this study confirm the original hypothesis
regarding the relevant presence and significance of mathematical concepts in Città. Most
significantly, they corroborate the essential aspect of the unexpected, as initially
recognized and stated by Gödel’s Theorem (also found in Lucretius and Galileo’s ideas).
An intriguing (yet not coincidental) implication has also been how mathematics and
literature commonly aim at triggering aesthetic response
In Città, Calvino uniquely applies mathematical concepts in such manner as to
enhance the agility of imagination while reading, by lightening the structure and
clarifying the view through the potential of favorably visual mathematical terms. These
concepts appear as a subtle net, a filigrana, which as examined, allows the possibility of
other spaces, found within and in between spaces, allowing a greater freedom for
creativity.
Through Calvino’s skillful writing, the conceptual and creative aspects of
mathematics and literature complement one another. This may occur in a parallel fashion,
or by intersecting each other, at times, even by following the same line; but always
looking for that point where reality and fantasy may or may not coincide, through
189
openings to new non-collapsing spaces that liberate creativity.” To find the point where
the imagined fortress does not coincide with the real one and then find it” (“Cybernetics
and Ghosts” 27).
The significance of this study relies in complementing previous studies at the
same opening other areas, spaces. The mathematical approach confirms the importance of
interdisciplinary connections between arts and science. In this case, the results were
particularly applicable between mathematics and literature. The impact of mathematics in
literature and vice-versa implies a reciprocal relationship, always present in a humanistic
tradition.
For don Quijote, mathematics is the most important science of all because it
encloses all the others, and its knowledge is essential to any caballero andante in every
step of his travel:
Es una ciencia –replicó don Quijote– que encierra en sí todas o las más
ciencias del mundo, a causa que el que la profesa ha de ser jurisperito y
saber las leyes de la justicia distributiva y conmutativa, [...] ha de ser
teólogo [...]; ha de ser médico; [...] ha de ser astrólogo, para conocer por
las estrellas cuántas horas son pasadas de la noche, y en qué parte y en qué
clima del mundo se halla; ha de saber las matemáticas, porque a cada
paso se le ofrecerá tener necesidad dellas (El Quijote II, 18).
The presence of intricate and subtle relationships between mathematical concepts
and literature suggests poetry of mathematics in Le città invisibili. Like a filigrana, which
may escape, be evasive, even invisible, never imposing, yet present, at times explicitly,
but most significantly within the crevices, the folds, seeming to appear and disappear,
manifesting deep repercussions. It is with this humanistic vision enclosing arts and
science that we may be able to attain creative freedom, to visualize the invisible.
190
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