Polytropy

Tophane-i Amire, 2017.03.25

Last week I wrote about the Turkish Impressionist Feyhaman Duran, born in 1886. Now my subject is the Hungarian-French Op Artist born twenty years later as Győző Vásárhelyi. His “Rétrospective en Turquie” is at the Tophane-i Amire Culture and Art Center in an Ottoman cannon foundry.

I fancy that Victor Vasarely is one of the artists I understand the best, though until now I have probably known his work only through books.

Victor Vasarely, Kodd, 1965
(Collection d’Art Renault, detail)

I have no clear list of favorite artists, if “artist” is taken in the narrower sense of painter or visual artist. I have a list of writers and philosophers: Charlotte Brontë, Henry David Thoreau, Somerset Maugham, Robin Collingwood, Mary Midgley, and Robert Pirsig. I have their books on my shelves; I can read those books at will; I could be content to read only those books. One can normally carry the works of a visual artist only in one’s head. As I think of the artists whose works I carry most closely, Dutch persons come to mind: Rembrandt, Van Gogh, Mondrian, Escher. Of these, perhaps Mondrian is most akin to Vasarely. I would rather have a Mondrian on my wall than a Vasarely. But I think I understand what Vasarely is doing, better than I do Mondrian.

Victor Vasarely, Étude sur carton,
Série « L’alphabet plastique, » 1960–75
(Aix-en-Provence, Fondation Vasarely)

I think I understand, because I fancy I used to do similar work.

printfreegraphpaper.com
pdf2jpg.net

One Saturday morning in the spring of my freshman year of high school, I woke before dawn, and I felt like getting up. I wake spontaneously before dawn every day now; but back then it was unusual.

1980.04.05

I must already have been thinking about how to create certain repetitive visual effects. That morning, I understood that if I drew three similar overlapping triangular arrays, each more widely spaced than the next, then by connecting vertices across the arrays, I could achieve the appearance of an array of cubes in perspective, the cubes joined only along their edges, three faces of each cube exposed to view.

Hagia Sophia, 2013.04.13

My work of sketching the cubes was repetitive, and it may have been tedious. I had help from the homemade light table that my godfather had given me: an old unpainted table with a drawer, in the top of which he had cut a rectangular opening, not much larger than a sheet of letter paper. A pane of frosted glass fit into the opening. Below the glass, screwed to the bottom of the drawer, was an array of three small fluorescent lights.

printfreegraphpaper.com
pdf2jpg.net

My “Uncle” Derry had used the light table to provide guidelines for his calligraphy. Some years before giving me the table, he had also given me an italic pen and instruction book, and I did learn the calligraphy taught in the book. I did not maintain the practice. I did not have any words, my own or somebody else’s, that I felt deserving of the labor of calligraphic treatment. But when Derry offered me the light table, I took it. And then, one spring morning before dawn, I used the table so that my dots on a sheet of graph paper would show through the plain paper on which I wanted to draw my cubes.

My depiction of 3y = 4/(x–x²), possibly around 1982

I did not have what is apparently called isometric graph paper. I did not know it existed. I had only rectangular graph paper, which I used in those years for various repetitive activities. When I was in the seventh grade, I sketched the graphs of the trigonometric functions, using the table of values at the back of Analytic Geometry, a textbook from 1949 that my mother had used in college. Using the equations that I found in the body of the book, I sketched the conic sections, including hyperbolas with their asymptotes. I did not have the patience to read the book from the beginning. If I tried, I was surely put off by the opening words:

If A, B, and C are three points which are taken in that order on an infinite straight line, then in conformity with the principles of plane geometry we may write

(1)      AB + BC = AC.

For the purpose of analytic geometry it is convenient to have equation (1) valid regardless of the order of the points A, B, and C. The conventional way of accomplishing this is to select AB to mean the number of linear units between A and B, or the negative of that number, according as we associate with the segment AB the positive or the negative direction. With this understanding the segment AB is called a directed line segment.

It is a thrill to learn that a larger quantity can be subtracted from a smaller quantity. However, it is a thrill that came to humanity only in recent centuries. Other thrills may be more readily enticing than this one. To me, at the age of twelve, the words of Alfred Nelson, Karl Folley, and William Borgman did not constitute the mathematics that I wanted to read for pleasure. Probably I could accept, at least, the assertion that a line segment consisted of a number of linear units.

Nelson, Folley, and Borgman, Analytic Geometry
(New York: Ronald Press Company, 1949)

René Descartes is credited with inventing analytic geometry, and he had no notion of a “directed line segment.” He wanted to solve such problems as had been described by Pappus of Alexandria in the fourth century. Why not start a modern course with some idea of such problems?

x = y² and xy = 2
from www.graphsketch.com

A good problem from some centuries before Pappus is to duplicate the cube, in the sense of making it twice as big: in modern terms, to find the cube root of two. A number of solutions were assembled by one Eutocius in his commentary on Archimedes. Either Eutocius or a scholiast of his manuscript was a student of Isidore of Miletus, compiler of the works of Archimedes, and one of the master builders of the Hagia Sophia, here in the former Constantinople. Among the cube-duplications that Eutocius offers, the one by Menaechmus involves conic sections, a standard topic of analytic geometry today. Menaechmus may even have invented conic sections for the purpose of duplicating the cube. In modern terms, the parabola given by x = y² and the hyperbola given by xy = 2 intersect at the points (0,0) and (∛4, ∛2), and so we have what we want.

Duplication of the cube by Menaechmus.
From Reviel Netz, The works of Archimedes: Translated into English, together with Eutocius’ commentaries, with commentary, and critical edition of the diagrams. Volume I: The Two Books On the Sphere and the Cylinder
(Cambridge University Press, 2004), p. 288

An analytic geometry text needs to make sense of all this. I tried to make this sense, in the analytic course that I taught in the spring of 2015. I hope I talked about Menaechmus on the first day, though I am not sure. I probably did bring in the model of a parabola, as cut from a cone, that I had constructed the previous fall. In the notes that I typed up during the course, I start with the abstraction of an equivalence class; but the examples given should be familiar:

• the fraction a/b as the equivalence class of ordered pairs (x,y) of counting numbers such that ay = bx (though this is not Euclid’s definition);

• the length of a line segment as the equivalence class of segments that are equal to it—equal in the sense of Euclid, for whom equality was what we call congruence.

My students had studied Book I of Euclid’s Elements the previous fall.

From “The Parabola,” October, 2014

I passed to conic sections as quickly as possible, less than a fifth of the way into what ultimately became my notes. First I wished to give a rigorous definition of proportion based entirely on Book I of the Elements. It is not till halfway through their own book that Nelson, Folley, and Borgman do reach conic sections. As with all other textbooks that I know, the authors are not rigorous. Perhaps they themselves never paused to consider the mystery of the so-called real numbers: how the geometric meaning of addition is obvious, but the meaning of multiplication is not. Descartes could make it seem obvious, because he had a theory of proportion, as developed in Euclid’s Elements. Today the analytical consequences of that theory are just taken for granted.

I set out here to talk about Victor Vasarely. I think I understand what he is about, because I too have been drawn to the activity of creating visual repetition. But I was so drawn mainly in adolescence. If the works in the catalogue of the Vasarely show at Tophane-i Amire are representative, Vasarely did not get serious about repetitiveness until he was almost as old as I am now.

Around the “five-dome saloon” (beş kubbe salonu) of Tophane-i Amire were quotations by Vasarely, in French and Turkish; they did not say much to me. The catalogue has a few essays about Vasarely, also in French and Turkish; I have read a few pages, and again they have not said much to me. A work of art ought to be able to speak for itself.

My pastel work, possibly as early as 1975

One must however speak the language of the work. Modern mathe­maticians may think they speak the language of Euclid, which is mathe­matics; but then they make mistakes, to the point of finding errors in Euclid that are not there. The problem is not that Euclid speaks ancient Greek, and moderns generally don’t. Reading Euclid’s original Greek might help the under­standing, if only by forcing the reader to slow down. I myself seem to have needed to read Euclid at the pace of first-year university students, in order to start seeing, as I imagine, what he really has to say.

Victor Vasarely, Autoportrait, 1940
(Collection particulière en dépôt à la Fondation Vasarely)

Vasarely made figurative works in his younger years. I myself did little of this when I was young. I painted abstractions from an early age. Apparently there are persons who don’t see the point of such things. My art teacher in the fourth, eighth, and twelve grades never saw the point, but fortunately Mr Larson never told me this at the time, unlike another teacher, who had been one of his teachers. I saw Sanfred Larson after college, when he lent me his copy of Collingwood’s Principles of Art. He confessed then to not having understood what I used to try to paint.

Victor Vasarely, Zèbres A, 1938
(Collection particulière en dépôt à la Fondation Vasarely)

Works like Vasarely’s zebras have an immediate appeal. Optical illusions are always interesting. Bright colors are pleasing. The show at Tophane-i Amire did seem to provide pleasure to those who visited.

Is there any deeper level to Vasarely? If there is, how would this be expressed, other than by the works themselves?

Ball-point pen, perhaps 1982

I understand from museum guards that visitors will ask of a painting, “How much is that worth?” People have been corrupted into thinking that everything has a definite value, which can be given as as number of dollars—just as, I think, our students have been corrupted, as I once was, into thinking that every line segment has a definite value, called its length, which can also be given as some number of units. Of some of my own works on paper, I have been asked, “What is that?”—as if it has to be something, in order to be worth looking at.

Victor Vasarely, Ambigu-B, 1970(Pecs, Musée Vasarely),
with reflected Renault logo (1972)

The abstract work of art would seem to me to be the the work that is least in need of explanation. Figurative works point to things outside themselves, and those things have meanings that may not be known to the viewer. Thus when in college I attended a lecture on Botticelli’s “Primavera,” I heard the lecturer explain who the personages were that could be seen in the painting.

I had the fantasy then that the painting ought to appreciable without any special knowledge—even, perhaps, without any familiarity with human beings. That does not make a lot of sense; but I don’t think I made a lot of sense in those days, at least about art. I asked the speaker what could be said about the “Primavera” if it were abstract. I don’t think he had a clear response; but what could he say? This was in Santa Fe, where there were artists and art patrons; another audience member told me “abstract” was “a technical term.”

1980.01.26

I had liked to paint in high school, even to the point of wondering if I ought to devote myself to doing only that. I quickly thought better of such a plan, since I could not imagine taking up the burden of justifying what I painted. When we were in the fourth grade, Mr Larson had asked us to write something about the meaning of art. I remember expressing several points, and I don’t know where they came from, or what they were, except for the last point, which was something like, “If a work of art is not accepted by the public, it is nothing.” I do not know if I considered then that an artist like Feyhaman Duran might paint purely for his own pleasure. Perhaps I could imagine the possibility, but thought it was not enough. Mathematics is something that one can do, and do correctly, without needing to consult any other person or even any thing out in nature. And yet one must be able to confirm with others one’s assessment of a mathematical proof. This makes it a virtual requirement that a mathematician’s work be somehow publicized. Then anybody who wishes can see what has been accomplished.

Victor Vasarely, Cheyt-Pyr, 1970–71 (Pecs, Musée Vasarely)

The Vasarely show at Tophane-i Amire has presumably shown something of what Vasarely accomplished before he died in 1997. To my mind, what he accomplished was the peace of mind that can come from following a pattern. But here I refer mainly to the pattern of a particular work of art. I followed such a pattern on a spring morning, thirty-seven years ago. I could possibly do the same thing now, but my interests have mostly moved on. Vasarely moved on to designing a Renault logo.

Note. Images here are by me unless I give links to their sources. I took the photographs of the Vasarely show at Tophane-i Amire, and of works therein, on March 25, 2017. The hexagonal arrays of small high windows made Tophane a great place for the show. Since I wasn’t carrying a dedicated camera, I used the camera of my old feature phone. Unattributed single works of art (if they be so considered) are by my younger self; I photographed them on March 26, using the same feature phone for consistency.