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AbStrAct sky would later call his geometry (which today is called called is today (which geometry his call later would sky fi confl and doubt without postulate.h Not famous Euclid’s Lobachev- ft icts, invalid consider they which in geometry, fithe developed(1802–1860) non-Euclidean examples of rst 1830 Between and 1850, Nikolai century. Lobachevsky (1792–1856) and János nineteenth Bolyai the of half second the enduring this intellectual scandal in exemplary fashion [1]. fashion exemplary in scandal intellectual this enduring do, mathematicians as feeling visionary such having of ity not be put right, concluding that there was no other possibil- in the fundamentals of the whole thing that absolutely could existence, the mathematicians came upon something wrong everything had been brought into the most beautiful kind of ventures of human existence,” wrote Musil,ad- adding that intense aft and entertaining er most the of some compasses en- mathematics that here precisely is it whereas lives, our of business serious the from far out themselves play which the few that remain today. . . . But these are harmless whims, science. this of face real the sees one itself mathematics within but utility, possible its toward not looks one when Only case. tellectual apparatus whose task is to anticipate every possible Mathematical Man,” writing that mathematics is an ideal in- “Th essay short e his in mathematics of role flthe on ected century,twentieth the ofMusilAt re-beginning Robert the ScAndAl intellectUAl activity.and usedthroughoutallhisartistic inspiration thattheMexicansculptorSebastianinvented,rediscovered of thismessageliesintheinfi ofmathematical nite varietyofforms realm offreedomandimagination,abstractionrigor. Anexample andspacecanbethe the earlytwentiethcenturiesisthatgeometry A strongmessagethatmathematicsrevealedinthelatenineteenthand E L Sebastian’s C SpaceandForms I T R A L A R E N E G 22 SS https://doi.org/10.1162/leon_a_01633 ©2020 ISAST with this issue. See www.mitpressjournals.org/toc/leon/53/2 for supplementary fi les associated Email: [email protected]. Istituto Veneto Scienze, Lettere Arti, IVSLA, Venezia, Italia. Michele Emmer (mathematician, fi lmmaker, writer, journalist, editorial adviser), Th e of one reason, pure of luxury bold the is “Mathematics scandal r e M M e e l e h c i M was that geometry mutated signifimutatedin cantly geometry that was of geometry. transformations became a codification ofthe diff erent types of classifigroups each the of Consequently cation mations. were invariant with respect to a particular group of transfor- fiof properties that the gures of study the as geometry bed space in the traditional sense but with sets of ordered Riemann, geometry had to deal not necessarily with points or varieties of any dimension in any kind of space. According to Riemann described a global vision of geometry as the study of tics through the general ideas of G.F.B. Riemann (1826–1866). incorporatedinto andbecameanintegral ofmathema-part fitheto unusualsort of curious eld, a and genre, until was it For some years non-Euclidean geometry remained marginal sense.commonstrong contrastwithsuch in was it because non-Euclidean hyperbolic geometry) “imaginary geometry,” cian Robert Osserman: cian Robert mathemati- the of words the Recalling science. abstract an Mathmaticspassé. is Euclidisthat abandonedor be should too rigid, too and axioms, that fascinates artists: Th e Euclideanscheme was combinationfantasy the itis between andofrules, freedom thatnoting worth Itis [3]. century twentieth the of ginning literatureandbe- art on the at had mathematical ideas new Th e diff oft it [and] . . . circumstances erent to clarity brings en power of universality, allowing a single rule to apply in very the ways.ofcarries itFirst,Abstraction variety a worksin Linda D. Henderson analyzed in detail the infl uencesthat artists [2]. and writers ofimagination the sparked andsphere public these new and exciting mathematical ideas fi ltered intothe lessregular and more geometricstartling fi gures. Many of istic culture and on art. Mathematicians began to produce human-onhave can ideas mathematicalthatpercussions re-profound the of examplesinteresting most the of one space—is of idea new the short, on)—in fourth the (from dimensions higher the ofand non-Euclidean of geometry It is important to mention that the discovery (or invention) erlangen Programerlangen classic ENRO o.5,N.2 p 5–5,22 151 LEONARDO, Vol. 53, No. 2, pp. 151–156, 2020 . However, this does not mean that it that mean not does this However, . of Felix Klein (1849–1925) descri- (1849–1925) Klein Felix of n -ples. what may be a confused situation. . . . [Another advantage] Sebastian’s Space and Forms is that it provides us with great freedom to let our imagina- From the point of view of a mathematician who looks at the tions roam, permitting us to devise new and alternative ver- links between culture and mathematics, it is of interest to see sions of reality—versions that may or may not correspond how the artist views and interprets some of the great ideas to something in the real world [4]. of geometry, of mathematics. A noteworthy demonstration A strong message that mathematics revealed in the late of the truth of this statement lies in the infinite variety of nineteenth and early twentieth centuries is that geometry forms of mathematical inspiration that the Mexican sculptor and space can be the realm of freedom and imagination, Sebastian has invented, rediscovered and used throughout all abstraction and rigor. Geometric objects and mathemati- his artistic activity (Fig. 1). cal ideas are of universal interest—and are also available to non-mathematicians, artists, writers and musicians—to be used, misinterpreted, mutated or distorted with their essen- tial impact and at the same time, for non-mathematicians, esoteric and mysterious: a search for an order but a victory of imagination. It seems a contradiction but it is not.

A Mathematical Approach to Art The history of the relationships between art and mathematics is a long and winding road. There was an increased interest in these relationships over the last two centuries, and Max Bill wrote intriguing words on this subject in his 1949 ar- ticle “The Mathematical Way of Thinking in the Visual Art of Our Time,” on the relationship between art, form and mathematics. He wrote about mathematics as the science of transformations, relations and connections. Bill wanted to point out that, while mathematical and artistic activity are obviously not interchangeable, the mathematical laws of space and of the relations between objects can and should be of great importance for art. Bill used words very similar to those mathematicians used. A real and clear definition of the way in which the relationships between the two disci- plines—considered by many to be so distant—should be is as follows: It must not be supposed that an art based on the principles of mathematics, such as I have just adumbrated, is in any sense the same thing as a plastic or pictorial interpreta- tion of the latter. Indeed, it employs virtually none of the resources implicit in the term “pure mathematics.” The art in question can, perhaps, best be defined as the building up of significant patterns from the ever-changing relations, rhythms and proportions of abstract forms, each one of which, having its own causality, is tantamount to a law unto itself. As such, it presents some analogy to mathematics itself [5]. [Previously:] By a mathematical approach to art it is hardly necessary to say I do not mean any fanciful ideas for turning out art by some ingenious system of ready reckon- ing with the aid of mathematical formulas. So far as com- position is concerned every former school of art can be said to have had a more or less mathematical basis. There are also many trends in modern art which rely on the same sort of empirical calculations. These, together with the artist’s own individual scales of value, are just part of the ordinary elementary principles of design for establishing the proper relationship between component volumes; that is to say for imparting harmony to the whole [6]. Fig. 1. Sebastian, Red Brancusite, iron and acrylic enamel, 1980. (© Fundación Sebastian)

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Héctor Tajonar was surely right when he wrote that the Having quoted and alluded to some of the significant links artist Sebastian “belongs to a tradition which goes back to between art and mathematics over the past centuries, ideas the beginning of western culture and continues to the pres- that were at least partially well known to Sebastian, it is use- ent.” He added: ful to look at some of the artist’s sculptures to highlight what has been the artist’s commitment (Fig. 2). Without forgetting Geometry has always been a fundamental tool for both what D’Arcy W. Thompson wrote in 1917, over 100 years ago: scientists and artists, to interpret the cosmic order. . . . Se- “We are apt to think of mathematical definitions as too strict bastian is a worthy successor of all of the above, having and rigid for common use, but their rigour is combined with been inspired by their knowledge, as well as by the Mö- all but endless freedom” [9]. bius band, combinatorial topology, Buckminster Fuller’s Similarly Enrique X. de Anda Alanís wrote: geodesic dome, and the postulates of Euclidean and non Euclidean geometry [7]. Though Sebastian used mathematical speculations as a source of invention . . . he constantly revises the harmony It is instructive to read the titles of Sebastian’s work of his world in accordance with proportional models that throughout the years [8]: Purple Dodecahedron, Rhombo- are already implicit in his visual language. A strange quirk hedrite, Hexaflexagon, Architecture, Structures, Transforma- in the dynamics of creativity: mathematical thinking con- tions, Variations, Golden Rectangle, White Binomial, From fronted by the freedom to invent forms. In my eyes, this is Order to Chaos, Band, Strange Attractor, Hypercube, Knot, what gives Sebastian his defining idiosyncrasy: geometry as well as some of the titles dedicated to important people: and mathematics are not limited to a pragmatic repetition Dürer, Kepler, Leonardo, Lobachevsky, Brancusi, Alberti, of formulae, but serve instead as keys to the labyrinth of Picasso, Descartes [8]. physics in nature; they are absolutely creative and, inas- It is clear just by reading these words and names that much as they are embedded in natural law, offer potentially significant facts and characters influenced and inspired Se- infinite resources [10]. bastian, first of all the evolution and mutation of the idea of space, starting from the classical Platonic forms, which The possibility of infinite possibilities becomes evident in remain vital even today, to the invention of topology, the looking at the works. In the early years the call to the regular- profound influence of Euclidean geometry, the Italian Re- ity of the Platonic solids seems to be a very strong reminder naissance, new nineteenth-century ideas of space and their to anchor his invention in a solid tradition, a long history influence on the historical avant-gardes and on modern and of shapes and modification of forms (Fig. 3). He could not contemporary art, not forgetting the ties with architecture avoid a strong reference to infinity, one of the great cultural and decorative arts, in a perennial need for a visible render- dilemmas not only in mathematics but also in art. He goes ing of the new shapes and surfaces that have been gradually from small to monumental dimensions to show the lesson of discovered (or invented?) by mathematicians. fractals, self-similarity, the infinite progression.

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_01633 by guest on 01 October 2021 To move into space, to invade and model shapes and so model space. Continuing to experi- ment with forms, with their variations, dimen- sions, colors. Acquiring a language of his own, axioms of the form, an alphabet, laws, based on an accurate use of the Ars Combinatoria, of the idea of variations. And possibilities are virtually unlimited, endless. A new geometry, a new universe of space with elements, with tools, with colors, with dimen- sions that vary, complementary to each other, and that fill space. In the end a visual lesson of geometry, mathematics, art, leading to more and different abstract results, where abstract means real and concrete, because they are visible and tangible, real natural objects of human creativity. A lesson on connections between the mate- rial and the abstract, between the concrete and intuition, between mathematics and reality, yet ensuring that none of these words, these ideas, lose their aura of mystery (Fig. 4). Of particular interest is topology, as de Anda Alanís points out again:

Fig. 3. Sebastian, Throne of Nezahualcoyotl, iron and acrylic enamel, 1974. From a scientific viewpoint, the analysis of ele- (© Fundación Sebastian) ments subjected to topological tension takes no account of their form, size or position; to resolve these problems, only order and conti- nuity are considered. . . . Topology is governed by rules, and in that regard has proceeded to a theoretical formulation of its geometric ex- pressions, a complex geometry far removed from traditional representation, not without logic, and with an enormous potential to feed fantasy. . . . Sebastian succeeded in building his transformable structures by mastering the laws of topology. This implies that he has un- derstood, in scientific terms, the dynamic of folding and then mastered a procedure which allowed him to discover or invent his own to- pology [11]. The bookWhat Is Mathematics? by mathemati- cians Richard Courant and Herbert Robbins also describes topology: Around the middle of the nineteenth century there began a completely new development in geometry that was soon to become one of the great forces in modern mathematics. The new subject, called analysis situ or topology, has as its object the study of the properties of geometrical figures that persist even when the figures are subjected to deformations so drastic that all their metric and projective properties are lost [12] (Figs 5, 6). Mathematician Henri Poincaré wrote the pa- Fig. 4. Sebastian, Monterey Gate, iron and acrylic enamel, 1977. per Analysis Situs (Topology) in 1895, the first (© Fundación Sebastian) publication on topology.

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Fig. 6. Sebastian, Thick and Thin in Three Movements, iron and acrylic enamel, 1982. (© Fundación Sebastian)

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_01633 by guest on 01 October 2021 A Very Old Story its artistic, symbolic and metaphysical meaning by the pre- Another thing to note: The titles of some Sebastian works Colombian Tairona civilization in what is now Colombia refer to words, deities and animals from ancient Mexican between 200 and 1600 CE. Many titles of Sebastian’s works civilizations before the arrival of the conquistadores. Thus, reference the cultures of the pre-Colombian Americas. these show the language of art that links his ancient cultural The history of human cultures is complex and fascinating, roots to new ideas about space, where the bond is the rigor full of surprises. Art and artists are sometimes able to render and freedom of geometry. visible the unimaginable, because as Shakespeare said: One must not forget that one of the first topological ob- There are more things in heaven and earth . . . jects, the Möbius band, in addition to having become an icon Than are dreamt of in your philosophy. of modern art under Max Bill, had already been created in —Hamlet

References and Notes 8 Héctor Tajonar, ed., Sebastian: Sculptor (Mexico City: Fundación Sebastian, 2009). 1 Robert Musil, “The Mathematical Man,” in Burton Puke and­David S. Luft, eds., trans., Robert Musil: Precision and Soul: Essays and 9 D’Arcy W. Thompson,On Growth and Form (Cambridge: Cambridge Addresses (Chicago: Univ. of Chicago Press, 1995) pp. 40–42. First Univ. Press, 1917; 2nd Ed., 1942) p. 1027. published as: Robert Musil, “Der mathematische Mensch,” Der Lose Vogel, No. 10–12 (April–June 1913). 10 Enrique X. de Anda Alanís, “What Would the World Look Like If I Were Travelling on a Beam of Light,” in Tajonar [8] p. 67. 2 M. Emmer, Visual Harmonies: Mathematical Models, in M. Emmer, 11 de Anda Alanís [10] pp. 72–73. ed., Imagine Math 3 (Berlin: Springer Verlag, 2015) pp. 43–68. 12 Richard Courant and Herbert Robbins, What Is Mathematics? An 3 Linda D. Henderson, The Fourth Dimension and Non-Euclidean Ge- Elementary Approach to Ideas and Methods (New York: Oxford Univ. ometry in Modern Art, Rev. Ed. (Cambridge, MA: MIT Press, 2013). Press, 1940) p. 353. 4 Robert Osserman, Poetry of the Universe: A Mathematical Explora- tion of the Cosmos (New York: Anchor Books, 1995) p. 145. Manuscript received 5 June 2017. 5 Max Bill, “The Mathematical Way of Thinking in the Visual Art of Our Time,” originally published in Werk 3 (Winterthur, 1949), Michele Emmer was full professor of mathematics at the reprinted with changes by the author in Michele Emmer, ed., The University of Rome “La Sapienza” until 2015, as well as a writer, Visual Mind: Art and Mathematics (Cambridge, MA: MIT Press, 1993) pp. 5–9, p. 8. journalist and filmmaker. He is a member of the Istituto Veneto Scienze Lettere Arti (IVSLA), Venice. His interests include PDE 6 Bill [5] p. 5. and minimal surfaces; relationships between mathematics and 7 Héctor Tajonar, “Introduction,” Héctor Tajonar, ed., Sebastian: the arts; and architecture, cinema and culture. He has been a Sculptor (Mexico City: Fundación Sebastian, 2009) pp. 13–14. member of the Leonardo Editorial Board since 1992.

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