Appeal of Symmetry

George Pólya (1887‒1985) was born György Pólya in Budapest into a converted Jewish family with deep roots in Hungary. Pólya’s economist father, Jakab Pólya, was elected corresponding member of the Hungarian Academy of Sciences in 1893. George Pólya attended an excellent high school, the Berzsenyi Gimnázium, in downtown Budapest, where his favorite subjects were biology and literature. When he enrolled at the University of Budapest, one of his five siblings, his elder brother, Jenő Pólya (1876‒1945), supported his studies financially. Jenő was a well-known surgeon, innovator in surgical procedures, and Professor of Medicine at the University of Budapest. He vanished during the Arrow-Cross (Hungarian Nazi) reign in 1944‒1945 and was probably shot into the river Danube.

Pólya tried and abandoned law, languages, and literature. When he showed interest in philosophy, his professor, Bernát Alexander, advised him to add physics and mathematics to his curriculum. Pólya, like many other outstanding mathematicians, was influenced by the legendary Leopold Fejér (1880–1959) of the University of Budapest. He wasn’t drawn by a shared research interest, but rather by Fejér’s recognition of the beauty and challenge of mathematics. Pólya spent a couple of years at the University of Vienna and then at the University of Göttingen, learning from great mathematicians. Further visits to Frankfurt and Paris added to his broadening world view of mathematics. On a 1913 Budapest visit, Pólya made the acquaintance of Gábor Szegő (1895‒1985), who became his lifelong friend and later his close colleague at Stanford University. Pólya’s next visit to Budapest was in 1967. In 1914, Pólya moved to Zürich, which was another center of great mathematics.

Pólya’s first appointment was Privatdozent (approximately, associate professor) at the Swiss Federal Institute of Technology (ETH) in Zürich, and he became full professor in 1928. In 1924, he was the first International Rockefeller Fellow for a year in England, and he received another Rockefeller Fellowship in 1933, which he spent at Princeton. By 1940, the looming Nazi danger became so frightening that Pólya and his wife decided to move from Zürich to the United States. Following other appointments, he became Professor of Mathematics at Stanford University. He retired officially in 1954, at the age of 67, but continued to teach until 1978, when he was 91. He was elected member of the National Academy of Sciences of the USA in 1976 and, in the same year, he was elected honorary member of the Hungarian Academy of Sciences.

Pólya started publishing papers in mathematics in 1912 and on the teaching of mathematics in 1919. It was characteristic of him to initiate new areas and contribute to traditional ones. His principal area was mathematical analysis, which deals with differentiation, integration, measure, infinite series, and analytic functions. Other areas where he contributed numerous papers included probability, combinatorics, algebra, and number theory.

In 1924, Pólya published a paper, Über die analogie der kristallsymmetrie in der ebene (On the analogy of crystal symmetry in the plane), in Zeitschrift für Kristallographie in which he enumerated the 17 plane symmetry groups [4]. The Russian crystallographer Evgraf S. Fedorov (1853−1919) was the first to describe the 17 two-dimensional space groups in 1890 or 1891. Pólya rediscovered them, as did others. What makes Pólya’s contribution stand out is that he provided a tiling for each of the 17 groups whose patterns did not contain gaps or overlaps. Pólya was the first who recognized the importance of these symmetries for art history and the decorative arts. He writes about this explicitly as if addressing his words directly to the graphic artist M. C. Escher ([4]; the actual quote is from p. 278): “Die bedeutung dieser symmetrien für kunstgeschichte und kunstgewerbe; es handelt sich nämlich dabei eigentlich um die symmetrie periodisch in der ebene ausgebreiteter ornamente, wie solche als stoff- und tapetenmuster, parkettierungen usw. jedermann geläufig sind. (The significance of these symmetries for art history and decorative arts; they are actually the symmetry of ornaments periodically distributed in the plane, such as those familiar to everyone as fabric and wallpaper patterns, parquetry, etc.)”

Indeed, Pólya’s study invited Escher’s curiosity as he was fast becoming interested in such patterns. When the mathematician and symmetry researcher Doris Schattschneider examined Escher’s archival papers in 1976, she found that the artist had copied by hand the full 1924 Pólya paper [5]. Schattschneider copied the paper in Escher’s handwriting and, in 1977, she sent it to Pólya, who was happy to receive it because he had lost his correspondence with Escher. Not only did Escher and Pólya correspond but the artist also sent Pólya a print of his Development I. It was made one month after he had read Pólya’s paper and was his first graphic work utilizing a tiling derived from his studies of Pólya’s paper [5].

In 2003, many years after Pólya’s death, two suitcases of his papers were discovered in the attic of his former home in Stanford. This is described in Dorris Schattschneider’s revised edition of M. C. Escher: Visions of Symmetry [6]. Hundreds of sketches and notes showed that he had seriously entertained the idea of writing a book, The symmetry of ornament. The closing sentence of his 1924 paper already refers to his plans ([4]; the actual quote is from p. 282): “Daß das mathematische studium der ornamente auch vom künstlerischen gesichtspunkte aus etwas interesse hat, will ich anderswo erörtern (That the mathematical study of ornaments also has some interest from an artistic point of view, I will discuss elsewhere)”. His book would have been aimed at artists and laymen, but he never produced it. As for his interest in crystallography, Polya’s 1924 paper was not an outlier either. Years later he published his counting theorem for the systematic derivation of structures [7]. This paper is referred to twice in the Historical atlas of crystallography [8], whereas his 1924 paper [4] is not mentioned.

Pólya had an exceptionally rich legacy in mathematical research, but his contribution to the teaching of mathematics was even more consequential. His focus was on problem solving. In 1924, he and Szegő co-authored a two-volume monograph in German followed by a revised and enlarged version in English in the 1970s [9]. His most famous book, How to solve it, was published by Princeton University Press in 1945 [10] after four other publishers had declined it. The book has sold well over one million copies and appeared in 17 translations. Pólya published two more unusual mathematical books, both in two volumes: in 1954, about plausible reasoning [11], and in 1962, about mathematical discovery [12].

Coxeter and Escher

Harold Scott MacDonald Coxeter (1907‒2003), who liked to be called Donald, was born in Kensington, in London, into an arts-oriented family. His father ran a business of surgical devices but as soon as he could afford it, retired and devoted his life to sculpting and singing. Coxeter’s mother painted. Coxeter’s early interest was in mathematics and music. He studied at Trinity College, Cambridge, receiving his BA in 1928 and doctorate in 1931. He was a Rockefeller Fellow at Princeton, where he met Hermann Weyl and other mathematics luminaries. Back at Trinity he attended Ludwig Wittgenstein’s seminars on the philosophy of mathematics. In 1936, Coxeter joined the University of Toronto and stayed there for his entire professional life. It was there that, in 1995, my wife and fellow structural chemist, Magdolna Hargittai, and I visited him, and recorded a conversation with him [3].

R. Buckminster Fuller dedicated his magnum opus, Synergetics, to Coxeter. Part of the dedication read [13]: “By virtue of his extraordinary life’s work in mathematics, Dr Coxeter is the geometer of our bestirring twentieth century, the spontaneously acclaimed terrestrial curator of the historical inventory of the science of pattern analysis. I dedicate this work with particular esteem for him and in thanks to all the geometers of all time whose importance to humanity he epitomizes.”

Coxeter’s writings were rich in interesting quotations that he did not collect in a systematic way; he just remembered them and included them in his actual writing. He was an avid reader who did not return to previous readings but kept moving to new books all the time. In his youth he was fascinated by the stories of H. G. Wells. Later, his interest shifted to the plays of G. Bernard Shaw. Coxeter keenly followed the developments in crystallography. He used to distinguish between what he called `crystallographic solids' and others, such as the icosahedron and dodecahedron. By the time we met in 1995, he realized that such a distinction had been blurred because of the discovery of quasicrystals.

He was one of our invited speakers at the Symmetry 2000 Symposium in Stockholm, which Torvard C. Laurent and I co-organized under the sponsorship of the Wenner-Gren Foundation. Coxeter was happy to renew his interactions with his crystallographer friends, such as Jack D. Dunitz and Alan L. Mackay, at the meeting. Coxeter’s presentation, The rhombic triacontahedron, introduced the talks of the symposium [14]. Coxeter also published fundamental books on geometry (e.g. [15-17]).

In the following I quote some excerpts from my conversation with Coxeter in which he talked about his interactions with Escher ([3]; the actual quotes are from pp. 38−39).

IH: You have had some connections with M. C. Escher.

DC: First, at one of the International Congresses of Mathematicians, which took place in Amsterdam, there was an exhibition by M. C. Escher. My wife, being Dutch, naturally talked to him when he was exhibiting his art to the mathematicians. So, she got to know him and that was very helpful; we kept up correspondence. Later I wrote an article for the Royal Society of Canada: my Presidential address for Section III, on symmetry. It included a Poincaré-style model of the tessellations of (30°, 45°, 90°) triangles filling the hyperbolic plane so as to form a black and white pattern. Escher saw this and thought it was just what he wanted. In some of his work he had got tired of filling the plane with congruent figures, fitting together, and he thought how nice it would be if they were not congruent but just similar and changed the size while keeping their shape. Escher liked these things because they fulfilled his wish to make a pattern in which he had fishes, for instance, of a good size near the center but getting smaller and smaller as he went towards the circumference. He made Circle Limit I, and then Circle Limits II, III, and IV. Circle Limit III was particularly interesting because it had four colors besides black and white. It was closely related to the hyperbolic reflection group that I’d described.

Did you inspire him to this work?

That’s right. He was very pleased with this idea. After he had seen that paper of mine, he did Circle Limits III and IV. He had done Circle Limits I and II before.

Did he construct his drawings with precision?

Extraordinarily well, yes. There was a very interesting apparent exception because in Circle Limit III, if you look at the rows of fishes following one another, they have white stripes along their backs so that the circle is filled with a pattern of white stripes that cross one another. It is remarkable that the spaces between the white arcs appear to form a tessellation of hexagons and squares. Yet the white arcs cross one another, three going through each vertex; therefore, they cross at angles of 60 degrees. In particular, you seem to have triangles all of whose angles are 60 degrees, and that, of course, is wrong because such a triangle would be Euclidean and not hyperbolic.

Bruno Ernst, in his book about Escher, The magic mirror, page 109, was similarly disturbed, saying, “In addition to arcs placed at right angles to the circumference (as they ought to be), there are also some arcs that are not so placed”. I was interested in this and looked at it for a long time, and at last I realized what had happened. By careful measurement, I saw that all those white arcs meet the circumference at an angle very close to 80 degrees instead of 90 degrees. In fact, each of the white arcs does not represent a straight line in the hyperbolic plane but one branch of an equidistant curve. When you put it that way, everything falls into place, and you see that Escher did those drawings with extraordinary accuracy; when I worked it out trigonometrically I found that the angle of 80 degrees is actually arccos [(2^¼ ‒ 2^‒¼)/2] ≈ 79°58’.

Was he aware of this?

Absolutely unaware. In his own words: “… all these strings of fish shoot up like rockets from the infinite distance at right angles from the boundary and fall back again whence they came”.

Was it intuition?

True intuition. He came to hear me give a lecture once, and I tried to make it as simple as possible; he said he didn’t understand a single word.

Mathematicians and crystallographers recognized Escher before anybody else. What was his main appeal?

It was the appeal of symmetry.

References

[1] Hargittai, I. (2021). MacGillavry–Escher–Mamedov and periodic patterns, IUCr Newsletter, Vol 29, No 1.

[2] Hargittai, I. & Hargittai, B. (2023). Brilliance in exile: the diaspora of Hungarian scientists from John von Neumann to Katalin Karikó, ch. "George Pólya", pp. 112-115. Budapest: Central European University Press.

[3] Hargittai, I. (1996). Lifelong symmetry: a conversation with H. S. M. Coxeter. The Mathematical Intelligencer 18, 35-41.

[4] Pólya, G. (1924). Über die analogie der kristallsymmetrie in der ebene. Z. Kristall. 60, 278-282.

[5] Hargittai, I. (1997). Transmitting M. C. Escher's symmetries: a conversation with Doris Schattschneider. Hyperspace (Japan) 6, 16-28.

[6] Schattschneider, D. (2004). M. C. Escher: visions of symmetry. New York: Harry N. Abrams.

[7] Pólya, G. (1937). Kombinatorische anzahlbestimmungen für gruppen, graphen und chemische verbindungen. Acta Math. 68, 145-253.

[8] Lima-de-Faria, J. (1990). Historical atlas of crystallography, pp. 16, 39; pp. 80, 89. Kluwer.

[9] Pólya, G. & Szegö, G. (1972, 1976). Problems and theorems in analysis, Vol. I and Vol. II. Springer.

[10] Pólya, G. (1945). How to solve it: a new aspect of mathematical method. Princeton University Press.

[11] Pólya, G. (1954). Mathematics and plausible reasoning, Vol. 1, Introduction and analogy in mathematics, Vol. 2, Patterns of plausible inference. Princeton University Press.

[12] Pólya, G. (1962). Mathematical discovery: on understanding, learning, and teaching problem solving, Vol. I and Vol. II. John Wiley & Sons.

[13] Fuller, R. B. & Applewhite, E. J. (1975). Synergetics: explorations in the geometry of thinking. New York: Macmillan Publishing Co., Inc.

[14] Coxeter, H. S. M. (2002). The rhombic triacontahedron, Symmetry 2000, Parts 1 and 2, edited by I. Hargittai &  T. C. Laurent, ch. 1, pp. 1-10. London: Portland Press.

[15] Coxeter, H. S. M. (1998). Non-Euclidean geometry, 6th ed. The Mathematical Association of America.

[16] Coxeter, H. S. M. (1989). Introduction to geometry, 2nd ed. John Wiley & Sons.

[17] Coxeter, H. S. M. (1973). Regular polytopes, 3rd ed. New York: Dover.