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 leave Magadan and return to the Ukrainian city of Lugansk,.where Timofei Grigor'evich worked until 1980 as laboratory director of 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 19505. 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 Mechanics- Mathematics 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.
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 Mechanics- Mathematics 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 stratifiedminimal 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-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 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 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. Poincaré 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." ***
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:
"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 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.'
"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 picture- photograph 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 animatedzfilm 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 createdonly 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 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."
* 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.
