Mathematical Impressions 0821801627, 9780821801628

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Mathematical Impressions

Mathematical Impressions Anatolii T. Fomenko with the writing assistance of Richard Lipkin

American Mathematical Society Providence, Rhode Island

Thomas J. Bata LIbr TRENT UNIVERSi

1980 Mathematics Subject Classification (1985 Revision). Primary 00-XX

Library of Congress Cataloging-in-Publication Data Fomenko, A. T. Mathematical impressions / Anatoly T. Fomenko p. cm. ISBN 0-8218-0162-7 (alk. paper) 1. Fomenko, A. T.-Catalogs. 2. Art-Mathematics-Catalogs. I. Title. NC137.F66A4 1991 741.973-dc20 90-47514 CIP

Copyright © 1990 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. °°

Dedicated to my parents, Valentina Polikarpovna and Timofei Grigor'evich Fomenko

Anatolii Fomenko: Ideas and Reminiscences by Richard Lipkin On the northern coast of the Sea of Okhotsk, along the Soviet Union’s frigid eastern shores, stands the port city of Magadan, a bleak industrial and mining center that marks the virtual edge of Soviet civilization. There, at the base of a vast mountain range, residents herd reindeer, repair ships, run fish canneries, and mine for gold along the tortuous Kolyma river. Today heavy industry dominates the sparsely populated area. And yet, Magadan has another history as well. During the rule of Joseph Stalin, Magadan served as the site of a notorious work camp, where many Soviet citizens lost their lives in labor to the growing Soviet empire. In the life of Anatolii Timofeevich Fomenko, Magadan played a critical role, for it was there that he and his parents spent nine hor¬ rid years, huddled together in virtual shacks through long Siberian winters, during the most formative period of his youth. And it was there that he first became aware of his talents as a visual artist and as a mathematician. Born on March 13, 1945 in the Ukrainian city of Donetsk, Fomen¬ ko is the only child of Timofei Grigor'evich Fomenko, a mining en¬ gineer, and Valentina Polikarpovna Markova, a teacher of Russian literature. The family was always extremely close, exuding a warm and tranquil atmosphere in which, through a combination of nature and nurture, Fomenko acquired from his father a taste for the natural sciences and from his mother a flair for humanistic studies. Early on, his talents for visualization began to emerge, culminating in his obsession with drawing and geometry. His mother, born in 1918, held extensive knowledge not only of Russian language and literature, but also of world history, literature, and painting. She helped to educate him in the visual arts, instilling in him a passion for art and cultivating his talents in drawing and painting, while his father imparted the wonder of studying the nat¬ ural world. Born in 1910, the elder Fomenko had descended from a long line of Dnieper Cossacks, whom the Russian Empress Katherine II had charged with the task of protecting Russia’s southern fron¬ tiers. Timofei Grigor'evich Fomenko graduated from the Donetsk College of Mines with a degree in mineral enrichment and went to work as laboratory director of the Donetsk Coal Institute before being sent to the country’s far eastern shores to serve as laboratory director of the Scientific Research Institute of Gold and Rare Metals in Magadan, from 1950 to 1959. Eventually, the family was able to

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leave Magadan and return to the Ukrainian city of Lugansk, where Timofei Grigor'evich worked until 1980 as laboratory director of the the Scientific Research Institute for the Enrichment of Coal. As Fomenko recalls, the family was hardly “unaffected” by the brutality of the Stalinist dictatorship and the “grave events” that the country experienced between the early 1930s and the late 1950s. To some degree, he adds, “these events left an inescapable imprint on the life of the family, but at the same time united it uncommonly into a small but harmonious collective.” Yet, despite the tremen¬ dous turmoil brought on the family by the nine years in Magadan, Fomenko’s own education did not suffer. Strong efforts were made to keep his learning broad and systematic, and as a result he remained at the top of his classes throughout his academic years. At age 13 he took first prize in the All-Union Children’s Literary Competition for a narrative on science and technology in the future, which was subse¬ quently published in the newspaper “Pioneer Truth.” And, even while submitting his writings to a local newspaper during his secondary school years, he finished his 11 year curriculum in 10 years, passed his final examinations without attending lectures, and received a gold medal upon graduation, an emblem of highest distinction. With his passion for mathematics fast emerging, he took first prizes in several mathematics and physics olympiads. In 1961, fol¬ lowing his victory in an olympiad organized by the Moscow Institute of Physics and Engineering, he received an invitation to study there. Ironically, though, the Institute’s medical commission required pro¬ fessional physicists to have good vision; it refused to admit Fomenko because of his nearsightedness. So instead he entered MechanicsMathematics Department of Moscow State University, where he has been ever since. Life and work at the University went well. In his third year, he began his first serious mathematical research, a problem in celestial mechanics, under the supervision of V. V. Rumyantsev. Yet, drawn by his powerful interest in visual and spatial problems, his attention moved increasingly toward the area of topology, which he took on almost exclusively in his fifth year. On the recommendation of P. S. Aleksandrov, founder of the Soviet School of topologists, he switched his academic focus from the mechanics division entirely to mathe¬ matics and worked under the guidance of P. K. Rashevskii on new methods for constructing “homogeneous totally geodesic models for generating elements of the cohomology rings and homotopy groups of symmetric spaces.” The results were successful. He was able to com¬ plete a classification of all those nonzero elements of the cohomology rings of compact symmetric spaces that realize totally geodesic sub¬ manifolds. In addition, his classification included all those nonzero elements of the rational homotopy groups of such spaces that realize totally geodesic spheres.

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Subsequently, Fomenko completed a three-year graduate program in two years. In 1969, he defended his dissertation on “Totally Geode¬ sic Models of the Cycles,” earning commendation for his research from the Academic Council of the Mechanics-Mathematics Depart¬ ment of Moscow State University, and, shortly thereafter, was offered a professorship. Soon after taking on his new position, he became absorbed with a particular mathematical problem posed by the Bel¬ gian physicist Joseph Plateau (1801-1883) and as a result turned his attention almost exclusively to a specific area of study, namely, the multidimensional calculus of variations and the theory of minimal surfaces. Indeed, the Plateau problem, so called, was formulated in the following way: prove the existence of a surface whose volume is minimal among all surfaces with the same fixed boundary. Fomenko was able to solve the multidimensional Plateau problem in terms of spectral bordism. In his paper, he presented a new method for constructing concrete minimal surfaces in symmetric spaces and a method for verifying that they are globally minimal. By 1972, he defended this work in his dissertation for MechanicsMathematics Department titled “Solutions of the multidimensional Plateau problem in the spectral bordism classes on Riemannian man¬ ifolds.” Two years later, he was awarded the Moscow Mathematical Society Prize for it. The work apparently caused quite a stir and sub¬ sequently he received an invitation from the International Mathemat¬ ical Union to address the International Congress of Mathematicians, held that year in Vancouver. During 1973 and 1974, Fomenko had also become fascinated with the use of computers to solve problems in geometry and topology. Working with mathematicians I. A. Volodin and V. E. Kuznetsov, he developed a new and unusually efficient algorithm for recognizing a standard three-dimensional sphere in the class of all three-dimen¬ sional manifolds given by so-called Heegaard diagrams of genus two. Meanwhile, still continuing to develop his theory of stratified minimal surfaces, he expanded his area of interest and began working in Hamiltonian mechanics and the integration of Hamiltonian systems of differential equations, drawing heavily from his previous experi¬ ences as a student of physics. Although this period of Fomenko’s life was one of intensive math¬ ematical research and creativity, he found time for romance. The warm and spirited Tatyana Nikolaevna Shchelokova entered his life, and in 1977 the two were married. A specialist also in algebraic topology, she teaches mathematics at the Moscow Institute of Steel and Alloys. As his output of mathematical research continued to increase, Fomenko found himself becoming even more interested in computers. For many years, he pushed forward the theme of computer geometry in his teaching at the University, especially in terms of computer-

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aided theoretical and applied investigations in geometry, topology, and the theory of differential equations. In fact by the mid 1980s, he and mathematician S. V. Matveev vigorously pursued computer geom¬ etry for solving classification problems in Hamiltonian mechanics and three-dimensional topology. In 1985, Fomenko applied topological methods to the problem of classification of integrable Hamiltonian systems of differential equations and, as a result, he constructed a new theory—the symplectic topology of integrable systems of equa¬ tions. For his work on this theory, the Presidium of the Academy of Sciences of the USSR awarded him a prize in 1987. To date, he has authored more than 140 scientific publications as well as 16 books and monographs. Yet, despite his intense efforts in research and teaching, Fomenko has managed throughout his life to merge mathematics with his other interests, especially art and music. Among other things, he has devoted considerable energies to working out what he calls “new empirico-statistical methods for analyzing narrative sources, such as historical texts, with the goal of detecting dependent and independent texts among a large collection of historical texts.” This problem arises, he says, “in the dating of historical texts and of the events described in them, and in the statistical analysis of ancient chronol¬ ogy.” In fact, he has written a book devoted exclusively to this subject. Much of his energies too have gone toward education in mathematics, and he has authored a wide variety of textbooks, especially in the area of modern geometry and topology. But the link between what Fomenko calls “mathematics and its visualization” is the theme that has figured so prominently in his life and work. Since the mid-1970s, he has created more than 280 images and at times has produced as many as 40 in a year. He sees himself as having drawn much of his creative energy from his mother during his childhood. Though not trained professionally, Valentina Fomenko was an active painter who strove to share her enthusiasm for the visual arts to her son. In fact, he describes her as being his first and most important teacher, a figure who imparted a certain “spiritual charge” that has long remained with him and influenced his life and work as an artist. As early as age 13, Fomenko recalls creating paintings and sculp¬ tures in which he tried to “represent the world of the ancient earth, a gaze into the past, before man had appeared on it, ancient land¬ scapes, animals, and so on.” These works attracted notice, were shown in Moscow at a formal exhibition, where he was awarded three bronze medals. Long fascinated with the works of medieval and Renaissance artists, he admired the creations of such figures as Leonardo da Vinci, Pieter Breughel, and Hieronymous Bosch, as well as Salvador Dali, Arnold Bocklin, Ciurlionis, and Vasiliev. While still a student at Moscow State University, he even had corresponded

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with M. C. Escher. For a time, he turned his attention to making, as he describes, “an independent study of features of the painting and drawing techniques of the old masters from the Renaissance and from the latter part of the Middle Ages, and, reworking this material, used it to create intuitive images reflecting the rich world of modern mathematics, the view from the inside, and its deep philosophy.” And yet despite his serious study of art, he does not see himself as an artist. “In my mind,” he says, “these are not just an artist’s images. In fact, I do not really think of myself as an artist. I am a mathematician. To me, my drawings are like photographs of some strange and interesting mathematical world. For me it is not important to be an artist but to represent images of this world so that others can appreciate it. To penetrate this world, you must study mathematics at a reasonably high level, maybe even be a professional mathematician. If you study mathematics only for technical purposes and do not stop to think deeply about the ideas, then you really cannot understand this world. In that sense I differ from other artists.” Fomenko’s first formal efforts at drawing began when he coau¬ thored and illustrated a book on topology during the late 1960s while still a student at Moscow State University. Containing some 40 illus¬ trations of mathematical images—“a view from the inside,” as he puts it—the book “Homotopy Topology” was a wide success in the Soviet Union and became a standard text for a whole generation of young mathematicians. No doubt much of the book’s popularity resulted from what Fomenko calls “an unexpected look at topology through the prism of drawings.” Subsequently, he pursued his production of drawings and paintings with extreme vigor, conceiving of new ways to picture mathematical concepts. Not only have his images filled pages of his own numerous books on geometry, but they have also been chosen to illustrate other books on such subjects as statistics, probability, and number theory. In addition, his works have found their way into the Soviet scientific and popular press. Many of Fomenko’s images reveal what he himself describes as “deep reflections about the essence of being and about the place of modern man—in particular, the learned man—in the stormy and unpredictable world surrounding him.” To a certain degree, one can perceive such feelings and thoughts directly in his images, especially in light of his earlier life in Magadan. At one end of his artis¬ tic spectrum are his purely mathematical images, which express a certain vision of forms and shapes, his interpretations of what he calls “deformations of space.” And yet, at the other end, are his allegorical images, which draw heavily on still life studies, works of the Renaissance masters, the Old and New Testaments, and his own life experiences—images of a vicious world that haunt his mind, a world in which he found himself a captive unwilling participant. These are images of pain, of suffering, of human beings undergoing

5

various “transformations,” seeking to free themselves of burdens they can barely endure. Deep within his subconscious mind, scenes from life, from math¬ ematics, from his imagination somehow meld together to generate visions he realizes on paper. In fact, many of his works are, as he says, “closely connected with music.” In 1963 at Moscow State University, he was one of the founders of a student musical club called Topaz to which he often refers when reflecting on his work as an artist. He perceives in himself a deep connection between music, mathematics, and art that originates in a certain “feel for the infinite.” “It seems to me,” he says, “that the basic motif which mathematics and music have in common is the ‘infinity motif.’ The professional mathematician is continually dealing with infinite processes, and this gives rise to a certain ‘feel for the infinite’ which in no way lends itself to formal description. Something similar happens in the world of music, a world which at first glance seems to be among the farthest from the ‘dry’ realm of mathematics. But the two worlds have in common a high level of abstraction. In both, an abstract symbol can generate an entire world of emotions. Poincare was right in assigning an important role to intuition. The genesis of a mathematical result and the genesis of a musical experience have something in common. Personally, I find that a melody suggests to me geometrical images (deforming surfaces, erupting topological objects, etc). I feel closest to the so-called late Romanticism of the late 19th-early 20th centuries. In music this was Bruckner, Mahler, Scriabin, Wagner.” * The presence of a romantic spirit is in many ways clear within Fomenko. And yet, the poetic urges to which he often refers are not separate from his passion for mathematics. “In no way do I consider my musical and artistic hobbies to be a rest from mathematics,” he says. “They are simply (for me) a somewhat different form of mathematical thought. My graphics, which have no formal connec¬ tion with mathematics, nevertheless bear the indelible imprint of my profession. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life. One does not put it aside.” ** As a result of seeing the world in this way, of trying to visualize and interpret the world as a mathematician, he stresses that he has always “been intrigued by the possibility of showing non¬ mathematicians the intrinsic richness of the mathematical world, whose charm can only really be appreciated after spending many years traveling along its fantastic landscapes.” *** : Quoted from Mathematics and the External World: An Interview with Prof. A. T. Fomenko, by Neal and Ann Koblitz, in: The Mathematical Intelligencer 8 (1986), No. 2, 8-17. ** ibid. *** ibid.

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Indeed, the mathematical world he sees and conveys is not just a fantasy, he insists, but another “very real world.” It is a world that he describes as visiting and one that he subsequently tries to share with his artistic audience. “I think of my drawings as if they were photographs of a strange but real world,” he says, “and the nature of this world, one of infinite objects and processes, is not well known. Clearly there is a connection between the mathematical world and the real world. This is the relationship I see between my drawing and mathematics. Because I am a mathematician, I notice things in the world I otherwise probably wouldn’t see.” The artistic reflections drawn from this world often come about as much after intense mathematical investigation as emotional reflec¬ tion. “In modern mathematics, there are two types of thinking, and I have difficulty deciding which one is better,” he says. “The first is when you start with an equation and the solution to some problem and then devise a geometrical interpretation. The second type is when, during some step of your investigation, you use some geometric intuition. In this case, the geometrical thinking, the shapes, help you to structure the equations. Of course this level of mathematical intuition is very informal, very flexible—some kind of clouds. You wander inside these clouds and then suddenly you see something. It is impossible to reconstruct the trajectory of thinking inside these clouds, but you see something, in geometrical terms. Then you must figure out what to do with the formulas. In my case, it is very helpful to have a geometrical picture, some kind of geometrical intu¬ ition. In my mind, I see geometrical pictures, sometimes complicated, sometimes flexible, sometimes clouded. I see deformations of space. Sometimes, in my dreams, I see mathematical images, and when I awake I see solutions to very complicated problems, as if the solutions are in some way contained in the geometrical shapes.” When Fomenko describes the content of his graphic works, he explains it in terms of two “layers,” a first mathematical layer and a second “philosophical and purely human” layer. In the so-called first layer, he explains, “almost all of the graphic works bear a mathematical content, revealing for the viewer concrete mathematical constructions, images, concepts, and theorems, or con¬ veying the atmosphere surrounding those mathematical ideas and their place in the contemporary science. They are intended to arouse in the viewer’s mind certain associations that play a major role in contemporary mathematics.” Of even more interest, though, is his “second layer,” where he states that the works bear a “definite natural-philosophical content.” They are, as such, “photographs made by him at times of his journeys in the parallel world of real ideas and reminiscences.” To explain how he moves into that special world, or state of mind, he offers the following description:

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“At first I am immersed in the ordinary world and am subjected to diverse external actions, including stimulus from professional ac¬ tivities, from books, from musical compositions, among other things. These external actions (among them some very mysterious actions) are sometimes superimposed so that their cumulative effects become especially strong. Then my thoughts (and this internal psychological state) begin to move in the direction of this special ‘parallel real world of ideas and reminiscences,’ access to which is usually barred and opens only at moments of strong emotional stimulation and times of vigorous mathematical investigations. “There is a kind of ‘blow’ on the frontier separating these two worlds, and I break through into a new world, leaving the world of the ordinary. In each case it is practically impossible to recall afterwards the causes that lead to this breakthrough. Apparently, this is not necessary, since the causes are particularly subjective and are of little interest to the outside observer. “Having fallen into the ‘parallel world,’ I begin a journey in it, sometimes a fairly long journey. Outwardly, this is expressed as almost complete isolation from the ordinary world; I do not react to ordinary actions. In some sense, such a state is like a trance or state of meditation well known to all who have been deeply interested in the foundations of ancient philosophy. “It is an immensely rich parallel world, absolutely unlike the ordinary world. Perceptions and movements in it are based not on the usual sense organs, but on intuition. Part of this world is made up of the ‘sphere of mathematics,’ something like the ideal sphere of Ideas in ancient Greek philosophy, where mathematical images, concepts, constructions, theorems actually materialize. Then, as I look over this virtual landscape, it is as if I photograph it with a ‘camera,’ making snapshots of the structures and events taking place. While there, I am able to glide through like a disembodied phantom, instantaneously carrying myself enormous distances, speeding into the cosmos and back again, moving forward and backward in time, penetrating objects, and drawing from a sea of information preserved in the memory of all that has happened. Although this parallel world coexists with the ‘real’ world, its laws differ from what are considered ordinary. It is possible to be simultaneously in several places, and one can observe the deep causal mechanism of events that people do not usually see. “The journey can last a long time. And as it continues, the ‘cam¬ era’ collects more and more information until finally it reaches some critical value. Then there is a sort of explosion. An artistic creation is born. And a work of graphic art emerges as a result, created while still in the parallel world, a process that amounts to little more than developing the photograph, so to speak. Sometimes the photograph

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turns out to reveal more than I can remember in the haste of taking it. Still, the image is not created but, as a photograph, per se, is developed and fixed. In fact, before starting to develop it, I often do not know what it will show, since I make an enormous number of photos during these journeys and cannot remember all of the various views or the details that fill each landscape. “From the point of view of the ordinary world, and from that of external observers, this means that, after beginning each drawing I never know what will eventually appear on the clean sheet of paper. At the beginning of the drawing, I do not know its end. Therefore I always work without rough copies, sketches, or outlines. The draw¬ ing, in a sense, appears all at once as a clean copy. If someone looks at the drawing while it is being made, then that person will see a blank sheet on which isolated marks appear. Each mark is final, and my hand does not return to it again. “In a sense, this way of drawing is like using a rag to wipe a thick layer of dust from a picture that already exists. At first, there is a solid gray surface from which the details emerge as it is being cleaned. Yet, each detail, as it appears, already has a finished quality, as if it were a real photograph. The process is very much like developing a picture. “At this point, having fully developed the photograph, I separate myself from the image, leave the parallel world, and return to the world of the ordinary. I look around and see the picture standing far away. I approach it with astonishment, an onlooker like all others, scarcely recognizing the image before me. With difficulty I can recall the journey, the picture awakening in me some vague recollections and associations. But not much more. Other viewers approach. They may ask: Where did this come from? What is the meaning of this detail? But I find it difficult to answer. I feel much like the outside viewers, examining the work in wonder. I try to reconstruct my memory of the lost original, understanding that the picture-photograph gives only a weak representation of that original, which still exists in the faraway parallel world of reminiscences made real. “Like an onlooker, my recollections of the original are in some sense like shadows fluttering on the wall of a cave, shadows cast by figures passing by outside in the sunlight, blocking the light at the entrance from time to time. Thus I can only comment on pictures in the following way, to suggest a series of associations constructed with the help of familiar images. With these associations, I try to arouse in a viewer’s mind a corresponding ‘resonance,’ and if that resonance succeeds, then the viewer, standing before the image, can in some sense feel the original, as it exists in its true state in the faraway ‘parallel world of ideas.’

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“Consequently, I assume the viewer will react reciprocally, trying to free his fantasy, allowing his associations to stew freely, hoping that in that person’s mind the lost original will suddenly flare up and that the physical picture before him will look only as a pallid reflection of that original.” “Of the many viewers who have seen my works, most say they awaken fantasies and force them to construct independent images in their own minds. The freeing of fantasies is especially important to mathematicians, given the way mathematical intuition interacts with the ordinary world. It is almost impossible to capture in words the true content of these works of art, since the process of creating pictures breaks down into so many steps. To present all the details systematically requires more space than would make sense, if it could be conveyed at all. In each case the original image is quite far from us, in the parallel world of ideas and reminiscences. The picturephotograph before the viewer is at most only the confused account of a wandering author, returning happily from a journey to a far land.” Fomenko insists that he has never regarded himself as a profes¬ sional artist. He is not a member of the Union of Artists of the USSR and has never been supported financially or otherwise for any of his artistic projects. And yet, he has been recognized as an artist as readily as a mathematician, his works being displayed in more than 100 exhibitions in the Soviet Union and abroad, including Holland, India, and much of eastern Europe. In fact, V. I. Tarasov of the Central Film Studio for Animated Films in Moscow produced an animated film in 1988 based on many of Fomenko’s images that has been shown repeatedly on Soviet Television. Commenting on one of Fomenko’s showings in the newspaper “So¬ viet Culture,” reviewer V. Shvarts wrote of his reactions and those of his colleagues: “With mathematicians everything was simple— they sought first and foremost mathematics in the works of the pro¬ fessor. It was a different matter for those in the humanities, who frequently saw in his drawings something of which he himself was not aware—echoes of fantastic romances and cosmic landscapes, poetic stanzas and human feelings. The eminent Soviet mathematician A. N. Kolmogorov said of these drawings that they could have been created only by a person who knows what an analytic function is and understands methods for constructing topological surfaces. Perhaps that is so, but I would like to add that [he is] also someone who has been able to look at ordinary things with the eyes of an artist. His works were born from precisely this fusion. Of course, Fomenko is first of all an academic, but an academic in whose spirit lives a poet. And certainly it is therefore that he had to achieve artistic creation. Professor Fomenko is one of those contemporaries of ours who are fascinated by life in all of its manifestations. These people

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are able to find what is most interesting in it and to share liberally with others.” A modest, pensive man, Fomenko conveys the presence of a seri¬ ous, thoughtful, philosophical soul. His gaze is intense. His words are carefully chosen. And his view of the world, meticulously thought out, comes through as clearly in his conversation as it does in his art. “My own impression,” he says, “is that the general laws of nature are so powerful that we can barely imagine their strength—and those laws rule our world. The trajectories of our lives, our motions in some sense, are determined by those laws, though we are free to move within fairly narrow channels. “As individuals, we are so small that we can see only a small part of this larger world, which is sufficiently bigger than our capacity to understand it. But through mathematics, we can get some general sense of what this larger world is like, though we certainly cannot understand all the details. That is simply impossible.”

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The Alexander horned sphere No. 4, 1967 (Topology, theory of embeddings of manifolds) India ink on paper, 22 x32.5 cm. This ribbed, stonelike convolution of space is an object known as the Alexander sphere, or horned sphere, fam iliar in the area of mathematics called the topology of manifolds. Here, a surface is contorted in such a way that, once topologically embedded in space, it partitions that space into two domains, which are non-simply-connected. As a result, an object forms with fingered hands that link without touching, a process that continues to infinity. At each step the number of fingers, or horns, that grow out of the shape doubles. The embedding of horns in space becomes more and more complex. In this image, consider a human figure, whose arms lead to hands that divide into fingers, endlessly dividing and intertwining with each other. 12

A space with nontrivial local homology No. 8, 1967 (Topology, homology theory) India ink on paper, 21.5 x32.5 cm. Reaching from in finity and pointing to another faraway infinity, a lanky hand finds itself coiled by a polyhedron, composed of an mfinite band of shells all glued together, each bonded to another at a single point, creating an infinite chain. Notice that to cut this infinite polyhedron in any one place, any arbitrary place, would mean cutting at least one shell, creating a hole. Such a hole is nontrivial. In fact, it is a one-dimensional cycle—a surface without boundary—that could not, for instance, sustain spanning by a film. Notice too that the base opening of each shell is sealed by the spiral of the next shell, a fact that gives this shape some of its peculiar and remarkable properties. Finally, see now that the shells decrease in size as they wind their way toward two singular poifits that designate the destination, the two infinities. 14

Topological zoo No. 11, 1967 (Geometry, topology, and minimal surfaces) India ink on paper, 21 x 32 cm. In this cavernous space, the gallery of a great austere castle, three beings watch from above as other creatures pass time in a menagerie of magnificent mathematical forms, each a different perturbation of physical space. Above and to the right, an animated polyhedron comes to life and begins to decompose itself, breaking down into its constituent parts, the scorpiondike shells of which it is made. Observe the tail of the seeming scorpion, arching upward and toward the shell’s head, revealing intuitively facets of the object’s structure and form. See how the shells ultimately come together to create a single, infinite polyhedron. Meanwhile, in the center of the vast hall, a large torus, or donut-shaped object, is turning itself inside out, transforming itself and the space around it. Interestingly, even though the torus, which has been cut, or punctured, twists in space and turns inside out, the new object is still a torus, although the inside and outside surfaces have interchanged. At lower left, ba thed in the shadow of a great pillar, lies an object called Antoine’s Necklace, quite familiar in topology. To its right, in the lighted area, rests a soap film, which stretches across a circular wire. Co?nposed by joining together an ordinary Mobius strip with a triple Mobius strip, this minimal surface is remarkable in that it can be contracted continuously along its boundary without tearing. It can even be transformed to create another object known to topologists as a punctured Bing house. Finally, in the room’s center, lies a large 2-adic solenoid. 16

A 2-adic solenoid No. 195, 1977 (Topology, differential equations, Hamiltonian mechanics, symplectic geometry) India ink on paper, 46.5 x57 cm. This 2-adic solenoid, a contortion of space, where surfaces fold in on themselves around a central axis, is known in topology as an object rich in complexity and unusual properties, useful in verifying many geometric conjectures. Though not obvious from its final appearance, this object comes about when many tori, or donut shaped objects, are embedded in each other. To build such a shape, one begins with a suigle torus, copies it, and then winds the copy around the original. This process is then repeated over and over, moving toward infinity. Eventually, one creates a set of tori inside of tori, unwrapped donuts twisting inside themselves, like snakes coiled into rings. And the process goes on to its very limit. In each case too, part of each torus has been stripped away, opening up its interior, to reveal the creation of this space. Mathematicians and physicists have become more interested in this shape in recent years as they find it arising in an area of study called Hamiltonian mechanics. These winding tori, so to speak, turn out to describe important characteristics of certain equa tions of m otion, particularly integrable differential equations and their solutions. So we have a case where a well understood, classical object in mathematics has been, in a sense, reborn, owing to recent discoveries in mathematical physics. 18

The star diagram of Hertzsprung and Russell No. 32, 1967 (Astronomy) India ink on paper, 21.5 x 32 cm. In the vastness of space, stars span a great void. For many years, astronomers have wondered how stars distribute themselves in the cosmos, how they clump together to form some larger arrangement, how their energy levels affect their arrangements in space. This image is derived from an astronomical chart called a spectrum-luminosity diagram. Our own sun lies along the center of the curve represented in this image. Although the stars located on this so-called diffuse curve have many different qualities and characteristics, they do in fact share something in common, although strangely, it is hard to define what this commonness is. Some people believe the common features arise because these stars were formed in one region of the universe and are of roughly the same age and chemical composition. During the early part of the 20th century, two astronomers, the Dutchman Hertzsprung and the American Russell, both sought independently to determine whether there is a dependence between a star’s spectral class and its luminosity. In the diagram that now jointly bears their names, a horizontal axis marks the stars' spectral classes, while the vertical axis shows their luminosity. Each star too is represented by a point. The stars do not fall randomly, or chaotically, on the diagram, but are concentrated along several lines. In this image, certain stars are highlighted, the so-called principal sequence of stars and subdwarfs. 20

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Geometry of the spectrum-luminosity diagram No. 30, 1967 (Algebraic topology) India ink on paper, 21.5 x32 cm. Imagine that you could take the stars of the universe and arrange them according to some feature or quality such as their brightness, or intensity, or some other set of features. Here is another interpretation of the Hertzsprung-Russell diagram, where the spectral class of many stars is compared to their luminosity. In this image, it is as if we were to envision the core of this diagram from one side, as if we were looking from the perspective of a region of stars of the large spectral class. Into infinity soar such stars as the roasting white dwarfs, the subdwarfs, the white giants and the supergiants, while superimposed on the picture lies an infinite polyhedral partition, which takes on the appearance of a coordinate grid. 22

A spectral sequence No. 29, 1967 (Algebraic topology) India ink on paper, 21.5 x 32 cm. In this grand region of space, long arrays reach out toward infinity, each the product of many cells where, in one case, a lone being cries out against the cold light before it. In the deep distance, hovering above, a mysterious surface arcs in, defining the space and warping our view of the pa rtitioned plane. In the field of algebraic topology, where physical spaces are manipulated symbolically, there is a technique known as the method of spectral sequences that is often used to compute the homology and cohomology, or structural likeness, of certain kinds of spaces. To do this, one breaks down that space into fiberings, a kind of unit, which are then computed algebraically in terms of an infinite sequence of arrays. In fact, each of these arrays becomes a term of the spectral sequence, and together they become connected by certain types of differen tial opera tions. In this picture, we see one such array, demonstrated under certain conditions. It is infinite and divided into cells, and each cell contains a certain group, most often an Abelian group. As such, we can actually understand this space not just in some vague and intuitive way, but in terms of a collection of algebraic facts that characterize the array. Strangely, by filling each cell of the array with specific information, and then processing that information by way of algebraic rules, we can obtain a much deeper comprehension of that space, of its structure. 24

The cylinder of a continuous mapping No. 51, 1967 (Homotopic topology) India ink and gouache on paper, 30.5 x43 cm. From deep within the pockets and caverns of a vast and far-off space, bat-like creatures soar together, spreading their wings, traversing the atmosphere, winding their way through a forest of cylindrical shapes that rise from a surface representing Earth. The soaring creatures journey past, exploring, navigating, and in some sense investigatuig the region they must traverse. But the structure of this space has its own story. It is built around vertical cylinders that carve up and define the region. They are cylinders of a continuous mapping, often used in homotopy topology. They are created by attaching each cylinder's base to a space with a mapping function. As a result, each cylinder grows larger and is represented as a truncated cone on the horizon. Shown too are infinite pillars, which represent orbits of specific translations in Euclidean space. 26

A fiber space No. 64, 1970 (Topology of manifolds) India ink on paper, 30.5 x43 cm. In this somewhat more hard-edged elucidation of space, planes, tangents, and fibers spin out from a central point. The space is fleshed out in a very angular way, as large fibers arc up and out, toward some far away destination on the horizon. Indeed, the central theme here is the fibering of a tangent to a circle with a single angular point. The circle, embedded in a two-dimensional plane, serves as the base of the fiberings, while other fibers spin off to make up tangents. The picture itself is a conditional representation of this entire process. Tangents have been twisted through angles, while one of those tangents stands out from the rest. Since the tangent is not defined at a singular point, the fibers in the picture grow smaller as they reach toward infinity. The circle, too, is represented as a drop, hanging by a singular point. Meanwhile, the space itself is organized by the shapes, wa rped by the presence of these strong, arching forms. 28

Spectral sequences and orbits of the action of groups No. 26, 1967 (Geometry) India ink on paper, 21.5 x32 cm. These rotating hands, almost like spectral lines themselves cast onto warped planes, mold the space that surrounds them. Indeed, they are symmetry groups, well known in complex physical problems. These groups can be discrete, or continuous. Here, the space is fibered into orbits of the action of a group, an orbit being a set of points derived from a single point on which the elements of a certain transformation group are acting. In general, each point has its own orbit, and each orbit has a certain volume and dimension. In this image, several families of orbits correspond to the actions of certa in Lie groups. For example, in the lower right corner of the image, there are orbits creating concentric spheres, the result of a subgroup of a group of orthogonal transform a tions acting on a Euclidean space. Interesting too is the place of this object in the theory of minimal surfaces, which are invariant under the action of certain groups. If the action of the group is nice enough, then a fiber space arises whose topology can be studied with the help of spectral sequences. 30

The action of the fundamental group on the higher homotopy groups No. 16, 1967 (Algebraic and homotopic topology ) India ink on paper, 20 x 29.5 cm. Imagine a certain spheroid in space. Imagine it tethered to some point of itself by an elastic tube. Now consider a fundamental group, a homotopy invariant of a topological space. Its elements are the classes of homotopic paths, which can be continuously deformed into each other. In this image, what hooks the two spheroids together is nothing less than a mapping function, represented intuitively. The loop represents an element of the fundamental group. Growing out of one spheroid is a long, thin tube that slides along the loop, ending up at its own beginning, leaving each spheroid to be replaced by a new spheroid. Again, the essential idea is one of mapping. Different loops will determine different mappings. Once the mapping process has ended and all transformations have beeri completed, the original spheroid leaves its place and wanders freely in space, while still remaining attached by a tube to its point of origin. 32

Homotopy groups of spheres No. 253, 1971 (Algebraic topology and homotopy) Oil on art board, 50 x 70 cm. Beneath an eerie light, within this strange cosmic space, rays of an ancient sun illuminate a fantastic castle, poised on the precipice of a rock-hewn cliff. Above hovers a great sphere, collapsing in as space undulates and folds in around it. A mapping function is at work, and under its efforts a large sphere crumples inward into a fairly complicated structure, enveloping a smaller inner sphere with several layers. The visible structure represents an intuitive projection. Shown too are types of singularities, degeneracies, and folds that can arise from just such a mapping. The mathematical idea is one of computing the homotopy groups of spheres, formed by mapping one sphere onto another, and in this case a large sphere onto a smaller one. These homotopy groups are important invariants of topological spaces that aid us in distinguishing one space from another. 34

Deformation of the Riemann surface of an algebraic function No. 229, 1983 (Theory of algebraic functions) India ink on paper, 44 x 62 cm. Underlying this twisted deformation of space, where long tubes intertwine to weave a tortuous egg-like shape, is a certain threedimensional model. The model shows a deformation of a Rieman n surface of a special algebraic function, set in four-dimensional Euclidean space. This surface is also considered to be homeomorphic to a twodimensional sphere with one handle as well as a two-dimensional torus. In terms of the theory of algebraic functions, we can construct this kind of Riemann surface by taking two copies of a two-dimensional sphere, making two cuts on each, and then gluing the corresponding cuts together. What we obtain is a torus, or donut-like object, represented as two spheres joined together by two tube-like cylinders (which are shown in this image). A curious feature of this form is that if we deform the underlying function, a polynomial, such that its roots coalesce, then so too does the corresponding Riemann surface follow. Vanishing cycles appear, singular points arise, and the surface loses its smoothness, hi this image, two roots seek to coalesce into one, and, as a result, the upper sphere gets smaller while the lower one grows larger, a process somewhat visible through the object’s cut-away sections. 36

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Spines of 3-dimensional manifolds No. 99, 1972 (Theory of 3-dimensional mani folds) India ink and pencil on paper, 30.5 x43 cm. A corner of the universe. Stone bu ilding. Veined roof. Lost souls. Ribbed sky. Consider this a meditation on the spines of three-dimensional manifolds, spines that can in a sense be considered each manifold’s private code. If you cut a manifold and then contract it onto a twodimensional skeleton, you can virtually create a spine. Like the genes of life, the spine contains nearly all the critical information about the manifold itself. 60

An orbit of the action of an infinite group No. 170, 1975 (Group actions) India ink and pencil on paper, 31.5 x44 cm . Suspended slate steps span a great desert distance, flowing smoothly across a vast sandy plane, between flattened dunes, under the heat of a ferocious midday star. See here an object, one associated with the action of a group, an action given on a space where the elements of the group are homeomorphisms of the space. Consider too an orbit of some infinite subgroup of the given group. If we picture some domain in the space as a single diamond shape—a three-dimensional polyhedron—and apply to it successively all the elements of the infinite subgroup, the result is an infinite sequence of polyhedra duplicating one another. This we can call the orbit of the polyhedron under the action of the subgroup. If images of the polyhedron regularly cover the whole space, only intersecting at their boundaries, then we have a fundamental region of the given subgroup. Note too that not all of the polyhedra are identical. Each differs slightly from its neighbor. This is okay. Why? Because they are roughly similar to one another by way of a homeomorphism. 62

The boundary of polyhedra can be diminished when they are glued together. No. 130, 1973 (Theory of polyhedra) India ink and pencil on paper, 30 x 43 cm. An alien world. Creatures living in a stepped landscape, rounding out their existence with light, in rebellion against the hard-edged, angular, local world in which they feel suspended. Polyhedra form the body and structure of this image, existing in the space until they are cut and glued together along their boundaries. Imagine that two polyhedra with boundaries are given. They are identified, glued together, and the boundary is actually diminished. This intuitively obvious circumstance lies at the basis of an important theorem in homology theory, that finds application in the modern theory of minimal surfaces. The picture also illustrates the process of cutting the Riemann surface of an algebraic function into the sheets of its single-values. In this picture, the cut is actually the fissure. 64

A nontrivial knot in 3-dimensional space No. 144, 1974 (Knot theory) India ink on paper, 31.5 x 44 cm. A knot hovers in three-dimensional Euclidean space, dangling from a slim line like a spider preparing to quash its prey. But it is not just any knot. It is a nontrivial concrete knot, since it cannot be deformed by an isotopy into a standard circle embedded in a plane. Likewise, in topology, knot theory is truly nontrivial. That is so not only because of its inherent fascination and complexity, but also because of its incredible applications in biology and chemistry, where proteins and long polymer chains boil down to complex knots that can be understood and untangled. Given a certain knot, how can we figure out whether or not it is trivial? This is a terribly complicated problem, although an algorithm does exist to solve it. In some cases it is much easier to prove a knot's nontriviality. To do this, we look for the knot’s partial invariants to make computations.

Between two maxima there is always a saddle point. No. 48, 1968 (Functions on manifolds) India ink on paper, 21 x 32 cm. Columns of men, lined up in parallel, stand on a plane that reaches far back onto a tumultuous landscape. A warped stone pillar soars into a stormy and energetic sky, where clouds flow like whirling cream and light rebounds off the folds in the distant hills. In general, this picture illustrates a theorem, the so-called saddle-point principle, which states that a smooth Morse function defined on a connected manifold, with at least two local maximum points, must have a saddle point somewhere in the middle. You can think of the proof intuitively. If you join the two maxima with a rubber band and let it go, it will settle at a comfortable place somewhere in the middle, designating a saddle pomt. In this image, the rocky landscape corresponds to a graph of a function with four maxima, three of which give rise to two saddle points. The third is not visible, hidden behind the central rock. The image also illustrates a type of topological restructuring in the theory of iso-energetic surfaces of integrable Hamiltonian differential equations. 68

A 2-dimensional sphere in 3-dimensional space can be turned inside out. No. 233, 1985 (Topology of manifolds) India ink on paper, 34 x51 cm. These wild, convoluted curlicues of tubings, turned in on themselves and wrapped around each other, define some space above a ziggurat-like habitat, set on a reflective sea—of glass—on which humanoid creatures run, seeking shelter. The point of this image is to show one interpretation of an amazing transforma tion, the process of turn ing a sphere inside out. While, on the one hand, we cannot turn a standard circle embedded in a plane inside out with a smooth homotopy, we can turn inside out a twodimensional sphere in three-space, in a class of smooth immersions, which allow for self-intersectio?is but not breaks or loss of smoothness. In this picture the basic steps of this amazing deformation are laid out. Using a smooth homotopy, we can deform the two-dimensional sphere so as to locate it near a so-called Boy surface, an immersion of the twodimensional projective plane into three-dimensional space, represented by a very symmetric surface. Through a series of transformations, the sphere’s inner and outer surfaces will eventually change places. 70

Simplicial complexes No. 113, 1973 (Algebraic topology) India ink and pencil on paper, 30.5 x43 cm. On grainy, stepped pyramids rest massive stone monoliths, each a monument to the rigid, angular, arid hard-edged life in this isolated world, devoid of faces. Human beings stand aligned before the stones, not even seeing the nearby flat hills, which are splotched and spare, without even a weed. The sky above is as bland as the world below. Only the stones define the desolate space. Underlying this scene is an idea, the notion of a simplicial complex, a polyhedron, the sum of building blocks glued together in a piecewise, linear structure. There are tetrahedra, prisms, polytopes, and cones over polygons, all different from each other and yet all composed of rectilinear tetrahedra. To make computations, topology employs homeomorphic images of balls, or cells, that come together to form cell complexes. In this picture, coarse, consolidated blocks dominate the scene—blocks that would suffice for computing many applied problems. 72

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Polyhedra and simplicial chains No. 109, 1973 (Algebraic topology) India ink and pencil on paper, 31 x42.5 cm. In this grainy world of monolithic structures, an army of small beings stands aligned on a terrace, a virtual plateau. The space is wide, open, and hollow, almost an airless place filled with radiation bouncing off the slate-like surfaces and creating a chasm in which only a few beings can survive. An underlying notion here is of simplicial chains, or objects forined from elementary building blocks marked by numerical coefficients. We can add and subtract the chains as elements of an Abelian group, and by introducing boundary operators we can also compute boundary chains and define homology groups. Intuitively speaking, think of a chain as an arbitrary collection of blocks, bearing numerical markings. 74

The boundary operator No. 128, 1973 (Algebraic topology) India ink and pencil on paper, 30.5 x42.5 cm. In this zero-gravity arena, blocks remain suspended, though aligned. Their feeling of great heaviness, of powerful mass, is counterbalanced by their weightlessness and by the light and energy that they reflect, bouncing back into the abyss. Three-dimensional bodies, homeomorphic to parallelepipeds, have a boundary consisting of six rectangular faces. By gluing them together we diminish their total boundary, annihilating two rectangles each time we glue bodies together. Each gluing creates a three-dimensional body whose boundary is homeomorphic to a twodimensional sphere. This picture illustrates the difference between the concepts of geometric and algebraic boundaries. If several polyhedra come together to form another single polyhedron, then we can compute their boundaries and form an algebraic sum, which can be used to compute the boundary of the single, larger polyhedron. 76

The method of killing spaces in homotopic topology No. 44, 1968 (Homotopic topology) India ink and color pencil on paper, 31.5 x43.5 cm. A great stone ledge projects out into a tumultuous sea of vapor and flame, oozing and whirling in places, though calmer in others. On the ledge, endless rows of nameless souls hang their heads in prayer, searching for meaning and asking for redemption. Strangely, underlying this image is a method for computing topological invariants, or homotopy groups. The many successive levels of the long ledge, filled with unending series of human beings, represent homotopy groups, each one of which corresponds to the next killing space. Such killing spaces are a sequence of special spaces designed to study a certain object, say a polyhedron. If necessary, we can first try to compute the homology groups, then the homotopy groups, of the polyhedron. The first nontrivial homotopy group turns out to be isomorphic to a homology group. Subsequently, we can construct a new polyhedron whose homotopy groups coincide with those of the original space. The idea is that the first nontrivial homotopy group of the original polyhedron is killed. In the resulting space, we compute subsequeTit homology groups, following a certain algorithm and find the second homotopy group of the original polyhedron. The process continues on infinitely. 78

The theorem on the coincidence of simplicial and cellular homology No. 106, 1973 (Topology, homology theory) India ink and color pencil on paper, 30 x43 cm. In the darkness of night, undulating forms, seeming spirits, move elegantly through a space defined by rows of tall, hard-edged columns at various angles. Below, a figure remains, contemplating the spirits above. Indeed, this is a topological space, and if it is triangulated, or represented as a simplicial complex, then one can compute the simplicial homology groups. In this picture, we see a representation of an idea—to replace a simplicial homology with a cellular homology. It turns out that the idea of cellular homology is already imbedded in simplicial homology, since simplexes with the same numerical coefficients can be consolidated into larger blocks, which can in turn be made into cells. The process is simple enough in theory, though the computations are very complicated. To represent the most interesting spaces as simplicial complexes, we often need a terribly large number of simplexes. The more complicated the space, the more simplexes are needed. So in general, we might say that simplicial homology groups are easy to define, but hard to compute. In contrast, the cellular homology groups are hard to define, but easy to compute. In fact, for so-called nice spaces, one remarkable theorem points out that simplicial and cellular homology groups are isomorphic. 80

A system of shrinking neighborhoods No. 105, 1973 (Mathematical analysis, topology) India ink and color pencil on paper, 30.5 x42.5 cm. Beneath a hazy sun lies a puddle of tar, sprawled out on the desert floor, like the descendant of a mighty river gone dry. A bar t'ests in mid¬ stream. The heat beats down and all is still. Observe that many mathematical constructions use pairs of open sets, such as sets A and B, where B is contained in A along with its own closure. In the construction of a pa rtition of unity, for exa mple, corresponding to a certa in cover of a space with a system of open sets, we often see such pairs of sets. This image represents an open set and a contraction of it. The idea is that, given some open cover of a compact topological space, one can sh rink slightly each open set in this cover so that the resulting set lies entirely in the original, while the new shrunken sets form an open cover of the original space. Of course, by expanding or contracting such open sets, they can change significantly. 82

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Singularities of smooth functions No. 180, 1976 (Mathematical analysis and geometry, theory of singula rities) India ink and pencil on paper, 31.5 x44 cm. Spread over a truly surreal literary landscape, giant books blow in the wind, their pages twisting and turning in ways that paper and gravity do not ordinarily allow. A star shines crisply from above, although a hurricane is on the way. Within this image are several singularities which can arise from a smooth mapping. Singularities are very intriguing entities, arising in so many areas of mathematics and physics, ranging from celestial mechanics to optics to wave theory. In the theory of singularities of differentiable mappings, various types of singularities can be classified within the natural equivalences generated by regular changes of coordinates. For instance, in the right-hand pages of the books shown, we see objects that reflect the work of mappings. A point, a fold, and a gather, for instance, are the only stable singularities—ones not destroyed by small perturbations. 84

Simplicial spaces, cellular spaces, crystal and liquid No. 190, 1976 (General ideas in geometry and algebra) India ink and pencil on paper, 43 x 61.5 cm. On this landscape of fantastic tumult, a lone figure escapes beneath a turbulent sky. Great planar objects grow out of the horizon, while the sky is convoluted as if it were a sponge, composed of foam partitions. The theme of this image is cellular spaces, which figure largely in the field of topology and can be formed easily by gluing together elementary bricks. The mental pictu re of a cell complex is of something pliable, soft, amorphous, flexible, and even animated—something like a deformed clay sculpture. In the image’s upper right-hand corner, an enormous, strange crystal is evolving, one with a complicated symmetry group. Indeed, a branch of group theory is the classifica tion of crystal structures, and in this case we can clearly see just how complicated the intrinsic symmetry of a crystal lattice can be. 86

Rolling and sliding No. 188, 1976 (Nonholonomic mechanics) India ink and pencil on paper, 41 x 62 cm. On an icy plane, disks skid around aimlessly—some bouncing, some sliding, others colliding with each other. In fact when objects like records, ice skates, or other kinds of blades slide along, they demonstrate nonholonomic systems, or systems with nonholonomic constraints. A classical example: when we ice skate, the blades of our skates cannot slide sideways. Mathematically speaking, this means that a differential equation corresponding to a nonintegrable distribution serves as a constraint. This picture investigates the motions of sharpened disks as they slide along, creating a mechanical system as well as some truly nontrivial mathematical problems, hi this case, we can create a nonholonomic system from a holonomic system by attaching semicircular blades to a rigid body revolving in space, sliding along the inner surface of a rigid sphere, inside of which we place a spin fling top. Each blade like this imposes a nonholonomic constraint on the original system. 88

Combinatorial contraction No. 75, 1973 (Combinatorial topology) India ink, pencil, and oil on paper, 30.5 x43 cm. In this dark, angular world, spiky pyramids align on an infinitely deep wall, juxtaposed with organic, undulating shapes. Underlying this image is an intuitive representation, a certain view of an operation known as a combinatorial contraction of a polyhedron. If the polyhedron contains a simplex with a free face, a face to which no other simplex is attached, then we can remove this tetrahedron without changing its homotopic type. In other words, by pressing the free face into the tetrahedron, we actually eat it up, leaving only those faces that are hooked to other tetrahedra. In this image, we see frozen in time an intermediate moment of the combinatorial contraction, before all of the three-dimensional simplexes have gone away. Only the central region is free of them. The free faces of the tetrahedra form a boundary of the threedimensional polyhedra. In the topology of three-dimensional manifolds, this process often comes up when constructing spines of a ma nifold with a boundary. 90

Geometric fantasy on the theme of analytic functions No. 206, 1971 (Analytic functions and surfaces) Oil on art board, 50 x 70 cm. On this surreal sea, whirling with activity, great soap bubbles and films pock the horizon, waves flow in from the distance, and a funnel draws up into itself parts of the central scene. Conveyed here intuitively is the character of analytic functions and manifolds, corresponding to which we find smooth geometric shapes and some singular points that form elegant curves and surfaces. Diverse analytic images appear on the scene, all sorts of bending curves, warped surfaces, and singular points, including certain minimal surfaces, such as soap films and bubbles. Also cloaked within certain objects, surfaces of constant mean curvature boundaries of interfaces where two different physical media achieve a harmonious equilibrium. Curiously, drawing an analytic curve by hand turns out to be a very complicated matter. However hard you try, most often you end up drawing not an analytic or smooth curve, but a curve of finite order of smoothness. —

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Discrete groups generated by reflections No. 147, 1974 (Algebra, group theory, mathem atical theory of crystals) India ink and pencil on paper, 30 x43 cm. Nearly resembling an exploding star, or an intersection of swords, this system actually represents mirrors in space, the reflections of which form a discrete group. In fact this class of groups plays a key role in geometry and physics, since some very profound questions in number theory and the theory of forms on various fields hinge on groups generated by reflections. Since a unit normal vector uniquely determines each mirror, or hyperplane, we can formulate all the properties of the group generated by reflections in terms of the set of vectors normal to the mirrors. The groups generated by reflections in mirrors in a multidimensional Lobachevsky space are especially interesting. Underlying this image is a special variety of groups, known as Weyl groups. 94

Construction of complicated polyhedra from simple ones, I No. 76, 1972 (Topology) India ink and color pencil on paper, 31 x43 cm. Dark monoliths, composed of blocks, define this vast and chilly fu turistic space, where ambient light bounces off invisible surfaces. The heavy edifices, virtual pillars of stone, themselves seem to soak up all energy. Meanwhile, long chains of small beings drop quietly from a ledge, like lost souls wandering between walls in a world that does not even realize they are there. Here, simple components simplexes, cubes, blocks—come together to form more complicated topological spaces. —

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Construction of complicated polyhedra from simple ones, II No. 77, 1972 (Topology) India ink and color pencil on paper, 30 x43 cm. The ancient light of distant galaxies travels th rough space, visiting a medieval temple that stands suspended on a great pedestal. Simple building blocks make up this scheme, each one equipped with a piecewise linear structure and, in particular, has faces that are rectilinear poly topes. If two building blocks have linearly isomorph ic faces, then one can identify them with the help of linear homeomorphisms, or nonsingular linear mappings. Consequently, by gluing together the blocks along the faces, we can create much more complicated polyhedra, for instance, the mysterious temple pictured here. 98

Construction of complicated polyhedra from simple ones, III No. 78, 1972 (Topology) India ink and color pencil on paper, 31 x42.5 cm. From a small collection of components—some blocks, pyramids, and cubes—an entire city grows, mushrooming out to fill a great space. Light pokes through the darkness, creatures move about, and city life takes hold in the multifarious towering dwellings, which dwarf human beings with their fantastic massive structures. That such complexity can grow out of so many simple elements is the underlying theme here. Mathematically, a nontrivial idea inherent in this scene is the fact that a simple process underlies the visible complexity. Although all of the local gluing operations are linear, the results are essentially nonlinear. 100

Construction of complicated polyhedra from simple ones, IV No. 232, 1972 (Topology) India ink and color pencil on paper, 34 x 51 cm. A starry night, where energies from above beam through the heavens onto an undulating landscape. Again, simplicity builds on itself to form complexity. By gluing together smaller pieces or building blocks, we can create more complicated polyhedra. Although these visible forms appear fairly amorphous, they can be broken down into simple, linear components. 102

From chaos to order No. 184, 1976 (Probability theory) India ink and pencil on paper, 31.5 x44 cm. An individual sits alone on a plane, observing the horizon and contemplating a great crystalline structure mush rooming out of the earth in the distance. Spread out on the ground are millions of pins. Indeed, in probability theory, we often find that, where large numbers of events are concerned, apparent chaos settles into an appreciable order. In geometry, a problem in probability can involve, for example, the distribution of needles falling randomly on a plane. This image does not illustrate a particular theorem but instead expresses graphically a philosophical idea—a long and careful study of chaos can lead us to discover a h idden regularity. To a certain extent, the giant crystal on the horizon embodies an ideal principle, an aspiration that provides a motivating force for much of science. 104

Mathematical infinity No. 196, 1977 (Geometric fantasy on the theme of concepts in general mathema tics) India ink and pencil on paper, 48.5 x 69.5 cm. Thousands of faces in a crowd cry out, encircling a single ominous head. Indeed, this image reflects a mathematician’s meditations on infinity, a concept that accompanies many theories and appears in various guises in geometry, logic, number theory, and many other areas. Potential and actual infinity, paradoxes of logic, unsolvable problems, the continuum hypothesis and its diverse versions, constructive mathematics, intuitionism (in the spirit of Poincare)—all of these come to life via the existence of mathematical infinity, the study of which presen ts fascinating philosophical problems regarding knowledge of the world around us. As for people, a suitable homeomorphism can identify differen t human beings from a geometric point of view, beginning with a single ideal hero. All of this too recalls the many medieval artists who tried to reflect their interpretations of physical and moral infinites on canvasses devoted to the sufferings of Jesus Christ. 106

Interior and boundary points of a manifold and symmetric spaces No. 80, 1972 (Theory of manifolds) India ink and pencil on paper, 31 x 43 cm. A meditation on spheres. Stretched out along a deep flat surface, spheres are aligned in columns that reach hack toward infinity. At center, a sphere is undergoing a transformation, becoming a sculpture, on whose edge grows a creature with arms that reach toward the sky. Dark brightens into light, one form grows out of another, and together they create a magnificent scene of rounded shapes. Here, these objects highlight the difference between the points on the interior and the points on the boundary of a manifold. By gluing together standard Euclidean balls and half-balls, we are able to create a manifold with boundary. Indeed, the sculpture poised on the edge of the large half-ball prevents it from being invariant under rotations about its central axis. 108

Theory of oscillations and wave processes No. 95, 1970 (Mathematical physics, geometry of partial differential equations) India ink and pencil on paper, 30 x 42.5 cm. An explosion of waves, solid and undulating, rhythmically vibrating. From a central point, shifting toward the right, a sine wave propagates, tempered by finer harmonics. On the horizon, great tidal waves are collecting, building, and waiting to break, revealing small surges too that in recent years have created much interest for mathematical investigations, especially in the area of solitons and partial differential equations. Real ocean waves are terribly complicated, as are the tumultuous waves that mount into tornadoes and hurricanes in the atmosphere. To understand those waves today, scien tists m ust approximate them with computer models. 110

Impact No. 92, 1973 (Theoretical mechanics and the mechanics of continuous media) India ink and pencil on paper, 30.5 x43 cm. Asteroids besiege a tranquil place, soaring through the atmosphere, plummeting downward. One crashes through a surface, shattering the smooth structure, upsetting what was an otherwise calm space. In physics, the mechanisms of powerful impacts create fascinating study, especially when we try to figure in all the types of motion, which usually lead to exotic solutions. One special case that spawns fascinating mathematics involves the collision of rota ting balls, which leads to two observable effects. One is the law of reflection of ideal balls and the other is the rule of composition of angular velocity vectors. When a ball collides with a surface, cracks can form in the plane that will eventually cause it to shatter. In fact, the theory of how those cracks spread throughout the surface creates fascinating problems for mathematical modelling. 112

Billiards and ergodicity No. 67, 1973 (Dynamical systems and probability theory) India ink and pencil on paper, 30.5 x 42.5 cm. A field of balls, reaching out in all directions, with human beings sprawled on top of spheres in the foreground. Imagine that only one ball were racing furiously around the field, rebounding from edge to edge, while a strobe light was capturing it in different positions. Eventually, if the images were superimposed on one another, then a whole field would be filled by images of balls scattered about. That is the case here in this study of the billiard ball problem, keeping in mind the central operating rule governing the ball’s motion, that the angle of incidence equals the angle of reflection. This picture represents the successive positions of a ball in motion, crisscrossing its way throughout an expansive plane, creating an image that is dense in some areas, sparse in others, with occasional empty spaces. 114

The Poisson-Laplace theorem and the Plateau principles No. 150, 1975 (Calculus of variations) India ink and pencil on paper, 31.5 x 44 cm. Bubbles clinging together in space, appearing almost as if they were a cluster of cells, radia ting energy from within and reflecting light from without. They represen t just the sort of sh ape one gets when shaking a soapy solution into a foam. Adjacent bubbles form edges and together they create a fascinating spacial structure. According to one of the Plateau principles, only three soap film sheets can meet stably on a single edge. Only four singular edges can come together at a vertex. A peculiar feature of this kind of system is that the bubbles can be pierced by a wire and not break down. Instead the bubble envelops the wire and it becomes part of the whole system. 116

A heavy top drifting in space No. 74, 1973 (Hamiltonian mechanics, symplectic geometry) India ink, pencil, and oil on paper, 30.5 x42.5 cm. A rigid body rapidly spins in space, like a top on its axis. Thus a gyroscope is created, which has the remarkable property of maintaining its position in space and holding to a course of motion, despite the twists and turns of the vessel on which it is seated. For this reason, gyroscopes play a critical role in navigating aircrafts and rockets. To study their equations of motion, mathematicians appeal to the theory of differential equations, symplectic geometry, algebra, and topology. Of interest too is the case where the rotating object is not symmetrical and its motion is chaotic. It may somersault wildly in space, and to understand its motion involves some very intricate mathematics. 118

How does a drop of liquid tear loose ? No. 73, 1971 (Calculus of variations) India ink, pencil, and oil on paper, 30 x42.5 cm. As globules of liquid fall from the sky, a series of solid blocks cuts through space, while a person emerges from one of the falling liquid balls. The idea for this image emerged from the mathematics that describes how liquid droplets form and then fall away from a sui'face. As the drops hang, a neck forms, and eventually they drop away from the surface to which they were clinging. The whole process has a special life of its own, a sequence of events that can be understood and modelled mathematically. In fact, from a mathematical point of view, surfaces that bound hanging drops can create thin capillaries, which are in a sense glued together to form other more complicated and interesting shapes. 120

Branched coverings over a sphere No. 187, 1976 (Algebraic functions and topology) India ink, pencil, and oil on paper, 44 x 62 cm. A sphere falls through space, collecting around it a branched covering that is meant to resemble a rosebud, its petals joined together forming various folds and pleats. Indeed, a sphere is a simply connected space that coincides with its universal covering space. In addition, there are many branched coverings over a sphere, ones that arise naturally in the theory of algebraic functions. Yet, by projecting a branched covering onto a sphere can we create the form of a rosebud, which clings to the ball's smooth surface and gives it texture. 122

Anti-Durer From the cycle: Dialogue with authors of the 16th century No. 175, 1975 (Geometric fantasy on the theme of concepts in general mathematics and random number generators) India ink and pencil on paper, 44 x 62 cm. Two figures, draped in flowing robes, sit meditating in this mathematical fantasy land. An hourglass rests in the foreground, a landscape rolls off into the distance, and a bell hangs on a wall waiting to be struck. In fact, the engraving “Melancholia by the 16th-century artist Albrecht Diirer inspired this image. Since Diirer's time, so much has cha nged in our understanding of the universe, and our knowledge of mathematics has so vastly grown. Yet, basic truths remain the same. Many mathematical ideas wend their way through this picture. For instance, hanging on the wall in the upper right-hand corner is the number e, the first 121 digits unwinding counterclockwise from the center in a spiral square. Inscribed in the clapper of the hanging bell is a representation of a separatrix diagram of a critical point of index one for a smooth function defined in three-dimensional space. Even the clouds in the background were serendipitously inspired by mathematical forms. ”

124

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A stare No. 110, 1973 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 30.5 x 43 cm. In the glassy eyes of an observant bobcat, figures glisten in the moonlight. The cat stares intently ahead, musing on its observers. Its deep, hypnotic pupils evoke many thoughts far beyond the realm of numbers. One should think of this cat as Begemot, the companion ofWoland, a hero in Bulgakov’s novel The Master and Margarita. Although this image refers to no mathematical theme in particular, the nature of the lines and the composition does call forth mathematical ideas, especially the notion of ha rm ony and balance in form. 126

A retraction of a space onto a subspace of it No. 126, 1974 (Topology) India ink and pencil on paper, 30 x 42 cm. In this explosive, almost electric image, forms twist and ooze against hard-edged structures that define the space. Organic and nonorganic shapes collide, creating what appears to be a hodge-podge of objects, all before a set of watchful feline eyes. In fact a time order underlies this setting, the theme being a contraction of a topological space onto a closed subspace, filled with large parallelepipeds. This deformation involves folding one space in onto another. The results are various amorphous, flexible structures, which settle softly onto rigid polyhedra. 128

Spines of two 3-dimensional compact closed hyperbolic ma nifolds of smallest complexity No. 245, 1987 (Hamiltonian symplectic geometry, topology of 3-dimensional manifolds, hyperbolic geometry) India ink and pencil on paper, 30 x40 cm. Great plumes of smoke belch relentlessly from the stacks of factories. Off into the horizon they reach, lining strange geometric fields, overshadowed by wispy clouds, ending in undulating cliffs that drop to an oily sea. Indeed, the intertwined plumes are themselves the chief ma thematical objects of this picture. They represent two-dimensional neighborhoods of the one-dimensional skeletons in the spines of two three-dimensional manifolds. These two manifolds, quite remarkable in themselves, are the first examples of iso-energy surfaces on which an arbitrary Hamiltonian differential equation is not integrable in the class of smooth integrals. The complexity of the manifolds is equal to nine, which we can see by counting the number of vertices of multiplicity four in the two-dimensional spines shown. 130

Picture of a gas flowing around a ball No. 167, 1975 (Aerodynamics, hydrodynamics, and geometry) India ink and pencil on paper, 31.5 x 44 cm. In the middle of a seemingly turbulent sea, a giant ball h overs in space, while all around it waves curl, crest, and roll. Below, dwarfed by its shadows, people stand and watch. This image is, in a sense, a meditation on the idea of flow, of the way liquids and gases take on shapes as they flow around a rigid sphere. When the fluid moves slowly, it remains smooth. When it flows fast, turbulence rumbles its surface. To study such laminar or turbulent flows, mathematical models come quickly into play, especially ones involving Lie groups and Hamiltonian geometry. Interestingly, these equations make up a special case of more general multidimensional equations that describe the motion of a rigid body’s multidimensional analogues. Meanwhile, the sphere continues to hover, the waters continue to flow. 132

Cellular spaces No. 250, 1970 (Topology) Oil on art board, 50 x 70 cm. In this whirligig world of large shapes in a space, the landscape is molded by the forms that grow upon it. Whereas a whole is the sum of its parts, here the larger objects come about as clusters of soft, pliable cells, or smaller units of which they are made. Indeed, cellular complexes make up an important class of topological spaces, ones that can be gotten by gluing together balls homeomorphic to Euclidean balls of various dimensions. Once assembled, the resulting complex in a sense hardens and takes on a life of its own. Here, an infinite sequence of cells and a very complicated space fill the horizon. 134

Random processes in probability No. 234, 1985 (Probability theory) India ink and pencil on paper, 33.5 x 50 cm. Throughout the universe random processes occur, ones that shape the world and affect our lives. Yet what makes a process random? Is there a hidden, underlying order? To appreciate this notion, consider the following scenario. A monkey stands in infinite space, holding onto a die. On each face of that die is another smaller monkey, who also holds a die. And on each of the faces of those dice are still smaller monkeys, each of whom holds a die, and so on, ad infinitum. At some point one monkey decides to throw his die in a random way, arid soon they all follow. Suddenly the whole system is in motion, in fact, apparent chaotic motion. This idea, originally suggested by A. N. Shiryaev, serves as a visual model for random processes that play such a crucial role in probability theory. Since one cannot portray this scene literally, a threedimensional cubic lattice is created instead for the sake of simplicity. In the depths of the space, one can barely see remnants of the original undistorted lattice, while in the foreground chaos has overcome order. 136

Statistical fantasy No. 249, 1987 (Mathematical statistics) India ink and pencil on paper, 38 x 49 cm. Dice tumble randomly in space, colliding with one another and altering the region around them. Above, in an eerie sky, clouds drip into place, like pudding onto a table. Chaos and randomness again make up the underlying themes. The motif of dice, which have random markings on their respective faces, calls forth thoughts of many problems in mathematical statistics. In fact, one can consider these images to be photographs of a strange, powerful, and fantastic mathematical world one that exists, regardless of how we perceive it, according to its own special laws.

—

138

The remarkable numbers pi and e, II No. 243, 1986 (Number theory, mathematical statistics) India ink and pencil on paper, 32 x 44 cm. Constructed from great cubes seemingly marked with splotches of ink, a monumental building rises from the ground. On top sits an apparent penthouse, perhaps the destination of the creatures scaling the tower's sides. On the front of the structure, encoded in the colored circles, is the number pi, while on the side wall is the number e. Each square represents a digit, exhibited as a square spiral of colored circles. Together thousands of these squa res add up to crea te a chaotic, variegated carpet. 140

The remarkable numbers pi and e, I No. 242, 1986 (Number theory, mathematical statistics) India ink and pencil on paper, 32 x 44 cm. Looming over an austere urban landscape, a tower stands, assejnbled from cubes whose faces are pocked with round markings. Below stand a church and clocktower, surrounded by monoliths suspended in space. Above, on a platform, a woman contemplates a sculpture. On the tower's front wall is inscribed a decimal expansion of the number pi, while on the side wall is the number e, both displayed in the form of black disks filling squares, row after row. Incribed too in this image is a fractal, a closed subset of the plane whose dimension is expressed by a fraction, rather than an integer. 142

Geometry and probability No. 246, 1987 (General concepts in probability theory) India ink and pencil on paper, 38 x47.5 cm. In a field of dice that lie randomly about, a handful pop up like kernels of corn. On the horizon, mud is pouring and piling up into a mountainous form. This image is the last in a series of three works devoted to simulating probabilistic processes. In recent years, geometry and probability have grown together, creating an interesting union. It turns out that many classical geometric objects, such as curvature tensors and affine connections among others, have arisen rather naturally in probability theory. In the case of certain theorems, geometric language lends itself comfortably to their formulation. Indeed, an underlying theme of this image is the intrinsic unity of geometry and probability. 144

Gaussian distributions, I No. 230, 1984 (Probability theory) India ink and pencil on paper, 44 x 62 cm. In this explosive image, large iron bars soar through space, emanating from a central source and flying about in all directions. Many of the bars fall in lines, while others drop chaotically. Surrounding the focal point are large mounds that not only define but also shape the space in which these random processes occur. Indeed, the theme of this image is the idea of distribution, which takes so many forms in the world around us. From the standpoint of mathematics, and particularly probability theory, random variables are distributed according to the normal law of Gauss, which leads to a graph reminiscent of a hill, a visual theme played out in many ways in this image. 146

Unstructured chaos and geometry No. 185, 1976 ( Geometry) India ink on paper, 45 x 61.5 cm. A whirlwind of blocks and bars whips through space, as if a hurricane had picked them up and strewn them around. Bars spin and twist in the atmosphere, flying and eventually falling to earth. What brings about such tumult? In some cases, merely cutting a well structured polyhedron into elementary pieces and shifting them about can lead to such apparent chaos. Here, breaking up a polyhedron has created a chaotic cluster of parallelepipeds. To reconstruct the original polyhedron, each simple block must remember its neighbor, the one with which it shared a boundary inside the original structure. Very little memory is needed. If each block remembers only its immediate neighbors and yet knows nothing about the structure as a whole, a very complicated system can still come together with surprising ease and regularity. 148

Gaussian distributions, II No. 231, 1984 (Probability theory) India ink and pencil on paper, 31.5 x44 cm. From far above fall iron bars, beams, rails, and stone blocks, which drop in lines from a cloudless sky, accumulating in piles and creating mountains on the horizon. Other bars lie randomly on the ground. In the distance stands a church, while a lone person leans against a giant bar in the foreground. Again the underlying theme here is distribution, shown in terms of bell-shaped curves that model normal distributions. In the picture’s center, we see how the process begins, whereas toward the horizon, distant mountains grow as beams fall for a long time. These processes are steady and cumulative. Fascinating is the fact that random events can form such regular structures, and that mathematical models can give insight into their shapes. 150

Homotopy and the tearing off of a drop from, a hard surface No. 248, 1987 (Topology, hydrostatics, and geometric intuition) India ink and pencil on paper, 34 x 50 cm. Suspended in space, floating before a crystalline complex in the background, is a globule of viscous fluid, dripping from one of its appendages. As a whole, the image shows the very beginning of a drop fanning, as the heavy viscous liquid hangs from a hard surface. Shown too, the beginning of a homotopy of a geometric object, which begins as a rigid shape and then softens into a pliable, resilient form that begins to deform continuously. Such images play a critical role in certain areas of mathematics, since many deep proofs have their basis in visually intuitive ideas. Often a formal algebraic proof comes to mind only after one formulates a clear geometric picture of some complex problem. Thus geometric imaginings themselves come to have practical value often in nonobvious, sometimes mysterious ways. 152

Homotopy and a viscous liquid No. 247, 1987 (Topology) India ink and pencil on paper, 34 x49 cm. A pudding-like liquid pours out of the sky, dripping from above into an enormous space, where people run randomly about. The gooey batter pulls, beads, and folds in on itself, masking a crystalline structure at center while, on the scene’s edge, a chu rch is being washed off of its foundation. In this image, the underlying theme is homotopy, the way in which an object is continuously deformed without breaking. In the case of this viscous fluid, which flows from some unknown vessel, it reveals what are called homotopically invariant properties. Notice the way the same material can be stretched, strained, twisted, and squashed, pulled apart like taffy or pressed together like dough. Yet, its surface remains mostly smooth and the bonds of connection are not broken. 154

Level surfaces of functions No. 141, 1974 (Geometry of manifolds and singular points of smooth functions) India ink and pencil on paper, 31.5 x44 cm. Hanging in the sky, smooth forms spin and twist like weather patterns, as if a tornado is reaching down into a chasm in the earth, probing the many levels below. On the surface, bulbous sculptures dot the landscape. At the lowest depth, shoes line up along a strata of a subterranean horizon. This image seeks to depict graphically the successive level surfaces of a function defined for the entire three-dimensional space, a level surface being a set of points whose value is constant. Indeed, the topology of a level surface can change along with the value of a function as it crosses a critical value. Here, critical points of the function appear on the corresponding level surface. At some singular points, separatrix disks drop down from one level surface to another. To a large extent, the structuring of the various level surfaces determines the topology of the entire manifold. 156

Gradient descent No. 176, 1976 (Calculus of variations and geometry) India ink and pencil on paper, 31.5 x44 cm. A long beam protrudes into space, pierced by two slender pins on which hangs a tattered cloth. In the background, great tubular structures jut into the sky, melting away at bottom while masked in part from view by a tenuous haze. The theme behind this image involves an operation of the geometric calculus of va riations, namely the process of lowering a cycle along the integral path of a vector field. As it slides downward, the cycle eventually catches on the critical points of some function. In this image the pins represent the critical points, while the beam represeiits the body of a manifold. The further study of such cycles under gradient descent falls under finite-dimensional Morse theory and its diverse generalizations to the infinite dimensional case. 158

Portrait of my wife, Tanya No. 192, 1977 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 44 x 62 cm. Tanya stares at a simple landscape, where only a few objects lie, a second portrait hovering below, on the horizon. Light reflects off her softly featured face, highlighting too the wrinkled space that envelops her image, both above and below. Clearly this image is not motivated by mathematics, and yet even here some mathematical thoughts creep in. Consider the background relief the rippled two-dimensional surface that fills most of the image. The breaking lines, the singular poin ts, the branches in the surface all grow out of images of certain analytic functions, particularly a Riemann surface that winds off to the right. In a sense, the background is like a tapestry of mathematical forms, woven together into a landscape that becomes an intriguing setting for this otherwise gentle portrait. 160

Mathematical preciseness and the concept of a fuzzy set No. 204, 1979 (General mathematical concepts) Oil on art board, 49.5 x 70 cm. This picture, which vaguely resembles a relief map of a coastline, is in fact drawn from the idea of the fuzzy set. For the most part, classical mathematics deals with precisely formulated, clear concepts. Yet, in recent years, fuzzy or diffuse sets have become popular, especially in such subjects as geometry and topology, computation, and probability theory. At the center of this picture, shrouded in mist, is an algebraic surface with singular points of the beak type. This particular image is intended to convey some of the sensations that surround work with fuzzy sets. Based on these ideas, many nonstandard mathematical analyses have arisen, including probabilistic approaches to the proofs of certain theorems. 162

Singular points of vector fields and the boundary layer in the flow of a liquid around a rigid body No. 254, 1980 (Differential equations and mechanics) India ink and pencil on paper, 36 x 52 cm. Nearly resembling a melting honeycomb in space, this image in fact reflects the motions of a fluid as it flows over a bumpy, complex object. As the flow moves past, it seems to mold the space around the object, creating smooth surfaces in some parts and rmffled eddies in others. The most interesting physical actions take place at the so-called boundary layer, or region near the object’s edge where the fluid first meets up with it. With a vector field, we can model the fluid motion, showing flow lines along which particles move. At the singular points in the vector fields, interesting forms well up in the liquid, such as sources, sinks, arid saddle points. In one instance, where onrushing fluids run up against the object’s surface perpendicular to the lines of flow, a saddle point is formed, creating a shape that is visually as well as mathematically interesting. 164

The music club “Topaz” in the MechanicsMathematics Department of Moscow University No. 191, 1976 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 40 x 75 cm. Portraits of mathematicians and physicists the former directors of students' music club Topaz (Moscow University) —fill this scene . Scattered amid the busts of persons are various musical and mathematical images, woven together into a busy and complicated setting. In the background sits a radio, besides which hovers a phonograph needle. Toward the foreground, a broken record lies beside the tail of a figure’s dragging garment. Bottles, cups, glasses, a pipe, a shoe, are all strewn about, along with sundry other items. Meanwhile, seated to the right, the composer Anton Bruckner, whose musical works were a favorite of the club, casts a watchful eye over the entire scene. —

166

The apocalypse (Revelation) No. 194, 1977 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 48 x 69 cm. Wild horses charge through this bizarre scene, amid chaos and wreckage, in a world consumed by tumult and confusion. In a sense, this image is really a geometric fantasy prompted by powerful biblical descriptions of the Apocalypse. Figures are strewn about in various stages of transformation. Objects—a radio, a teapot, a chessboard have been cast around in the clutter. Even some mathematical ideas come into play, such as notions of infinity. Deformations of human figures call forth the idea of homotopy and homeomorphisms. Even the clouds in the sky recall fractal images. —

168

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The temptation of St. Anthony No. 226, 1979 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 61 x 85 cm. In this highly chaotic scene, human and animal figures move about, engaged in a wide variety of activities. A giant two-headed monkey races over the horizon beside a figure on horseback, who is carrying a scepter and galloping over a sea of drums. To the left, side by side in a long line, figures sound their trumpets into the sky, while below another figure plays a keyboard. Seated on stereo equipment, yet another figure reads a mathematics notebook, while a central figure puzzles over the entire setting arid smokes a cigarette. In a sense, this image was inspired by the medieval legend of the temptation of St. Anthony, combined with certain mathematical ideas and images. For instance, the trumpets in the upper left are based on funnelshaped surfaces on which a hyperbolic metric is realized. Infused in other parts of the image are subtler mathematical ideas. 170

Anti-Breughel From the cycle: Dialogue with authors of the 16th century No. 189, 1976 (Extra-mathematical associations in mathematical images) India ink and pencil on paper, 46 x 74 cm. This bizarre, fantastic, medievalstyled image is a meditation on “The Alchemist, ” a famous engraving by Pieter Breughel. It is intended to be an ensemble of images and ideas reflecting the evolution of scientific thought during the past 300 years. Although this image illustrates no theorem in particular, mathematical ideas arid images are laced throughout, giving rise to the image’s underlying structure. For instance, cups of molten metal disappearing over the horizon suggest the idea of mathematical infinity. The twists and folds of the background clouds hovering in the sky call forth certain analytic functions. Various deformations of human bodies rest on the concept of homeomorphisms and homotopy. Throughout, the motions of figures and objects recall the concept of dynamic flow. 172

Geometric fantasy No. 199, 1968 (Extra-mathematical assocations in mathematical images) India ink and color pencil on paper, 61 x 85 cm. A field of cloaked figures fills a seemingly cavernous space, where rocky plateaus define surrounding walls. Indeed, in this geometric fantasy, the ideas of mathematical infinity and transfinite numbers work their way into the structures depicted in the image. In a sense, each infinity takes on the characteristics of a new number. When one considers an in finite sequence of such numbers, a new infinity appears that can be treated as a number in the next hierarchy. This process is endless, since new infinities can be added and operated on just as if they were ordinary numbers. 174

Geometric fantasy No. 200, 1971 (Extra-mathematical associations in mathematical images) India ink and color pencil on paper, 61 x 85 cm. In this geometric fantasy, nonrigid forms dominate the landscape, which is itself an amalgam of waves and undulating shapes. On the horizon, clouds hang almost as if a forest fire were burning into the flight, while below, the sea seems to wash up against a shore, leaving bare a drifting sculpture that punctuates the central space. Among the mathematical ideas interwoven into this scene are singular points of algebraic functions, Fourier series, fractals, dynamical systems, strange attractors, wave fronts, periodic mappings, and orbits of the action of discrete groups. 176

Topological restructuring of level surfaces of smooth functions on manifolds No. 198, 1972 (Geometry) India ink and color pencil on paper, 61 x 85 cm. Windblown mountains and dunes fill this desert landscape, whose craggy peaks and valleys are highlighted by a distant source of illumination. Roughedged rocks come together in places to create virtual cliffs that form borders to otherwise still and lifeless flats, while soft clouds float gently throughout. From a mathematical point of view, each smooth function on a smooth manifold fibers the latter into level hyper surfaces. As the value of the function increases, for instance, the corresponding level surface is deformed inside the manifold, changing its position. When the function’s value passes through a critical point, the level surface changes its structure and topology. Thus neighboring level surfaces differ in that their critical points fall in the spaces between the hypersurfaces. In a sense, thematically, this picture reveals how level surfaces are restructured, especially where they are portrayed graphically as layers of clouds in the sky. 178

Index of Images The Alexander horned sphere

12

A space with nontrivial local homology

14

Topological zoo

16

A 2-adic solenoid

18

The star diagram of Hertzsprung and Russell

20

Geometry of the spectrum-luminosity diagram

22

A spectral sequence

24

The cylinder of a continuous mapping

26

A fiber space

28

Spectral sequences and orbits of the action of groups

30

The action of the fundamental group on the higher homotopy groups

32

Homotopy groups of spheres

34

Deformation of the Riemann surface of an algebraic function

36

Morse functions and the theorem about the Euler characteristic

38

The separatrix diagram of a critical saddle point of a smooth function on a 3-dimensional manifold

40

Proper Morse functions on 3-dimensional manifolds

42

Motion of a heavy rigid body in space

44

An algebraic Rummer surface and its singular points

46

2-dimensional polyhedra and incidence matrices

48

Simplicial, cubic, cellular chains

50

181

Algebraic surfaces of higher order and the simplicial approximation theorem

52

The Euclidean plane— the simplest minimal surface

54

A theorem in symplectic geometry

56

Turbulence and associations outside mathematics

58

Spines of 3-dimensional manifolds

60

An orbit of the action of an infinite group

62

The boundary of polyhedra can be diminished when they are glued together.

64

A nontrivial knot in 3-dimensional space

66

Between two maxima there is always a saddle point.

68

A 2-dimensional sphere in 3-dimensional space can be turned inside out.

70

Simplicial complexes

72

Polyhedra and simplicial chains

74

The boundary operator

76

The method of killing spaces in homotopic topology

78

The theorem on the coincidence of simplicial and cellular homology

80

A system of shrinking neighborhoods

82

Singularities of smooth functions

84

Simplicial spaces, cellular spaces, crystal and liquid

86

Rolling and sliding

88

Combinatorial contraction

90

Geometric fantasy on the theme of analytic functions

92

Discrete groups generated by reflections

94

182

Construction of complicated polyhedra from simple ones, I

96

Construction of complicated polyhedra from simple ones, II

98

Construction of complicated polyhedra from simple ones, III

100

Construction of complicated polyhedra from simple ones, IV

102

From chaos to order

104

Mathematical infinity

106

Interior and boundary points of a manifold and symmetric spaces

108

Theory of oscillations and wave processes

110

Impact

112

Billiards and ergodicity

114

The Poisson-Laplace theorem and the Plateau principles

116

A heavy top drifting in space

118

How does a drop of liquid tear loose?

120

Branched coverings over a sphere

122

Anti-Diirer. From the cycle: Dialogue with authors of the 16th century

124

A stare

126

A retraction of a space onto a subspace of it

128

Spines of two 3-dimensional compact closed hyperbolic manifolds of smallest complexity

130

Picture of a gas flowing around a ball

132

Cellular spaces

134

Random processes in probability

136

Statistical fantasy

138

183

The remarkable numbers pi and e, II

140

The remarkable numbers pi and e, I

142

Geometry and probability

144

Gaussian distributions, I

146

Unstructured chaos and geometry

148

Gaussian distributions, II

150

Homotopy and the tearing off of a drop from a hard surface

152

Homotopy and a viscous liquid

154

Level surfaces of functions

156

Gradient descent

158

Portrait of my wife, Tanya

160

Mathematical preciseness and the concept of a fuzzy set

162

Singular points of vector fields and the boundary layer in the flow of a liquid around a rigid body

164

The music club “Topaz” in the Mechanics-Mathematics Department of Moscow University

166

The apocalypse (Revelation)

168

The temptation of St. Anthony

170

Anti-Breughel. From the cycle: Dialogue with authors of the 16th century

172

Geometric fantasy

174

Geometric fantasy

176

Topological restructuring of level surfaces of smooth functions on manifolds

178

184

DATE DUE / PATE DE RETOUR

APR Af ‘R 1 5 199 4

CARR MCLEAN

38-297

ISBN 0-8218-0162-7