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M. C. Escher: Visions of Symmetry

Doris Schattschneider
Harry N. Abrams
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
BLL Committee
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Good news! Doris Schattschneider's classic M. C. Escher: Visions of Symmetry is back in print and is better than ever. Taken out of print by the original publisher, W. H. Freeman, it has been reissued by Harry N. Abrams, probably the foremost publisher of art books in the United States. As one would expect, the quality of the binding, the paper, and the color printing are all first rate. Nevertheless, it comes at a surprisingly modest price. The illustrations from the original edition, most of them in color, are all present here. The first edition, of 1990, was printed in the United States, this new edition in China. The color in this somewhat expanded edition is a bit softer and the paper slightly off-white and somewhat less glossy. The overall effect is, if anything, even more agreeable than that of the first. The softer paper is apparently thicker enough that the new edition appears to be heftier, but in fact only twenty or so pages have been added. There's a new one-page introduction by Douglas Hofstadter, and a new well-illustrated Afterword to bring the reader up-to-date on developments in Escher scholarship since 1990.

A lot has happened since then: there have been additional conferences on Escher, some events for the centennial of Escher's birth, and the opening of a new Escher museum in The Hague. I do not plan here to describe at length the content of the first edition since that is known to many readers. Briefly, the book illustrates, with mathematical commentary, Escher's work in which he so ingeniously used symmetry to get his amazing patterns in the plane. For mathematicians, the underlying mathematics is interesting — but I emphasize above the quality of reproduction of the art because ultimately, for most readers, it is not the mathematics that stimulates interest in Escher's work, it is his dazzling ingenuity and use of color in creating these unlikely configurations.

In the first edition Schattschneider described in some detail the 1924 article of George Pólya in the Zeitschrift für Kristallographie in which Pólya showed that there are 17 symmetry groups of periodic patterns in the plane (a fact known in 1891 to the Russian, E. S. Federov, but this escaped the attention of both Pólya and Escher). The patterns corresponding to the symmetries (illustrated with examples by Pólya in his 1924 paper) were studied and copied by Escher. Numerous examples are given in the text. When the original book was written Schattschneider knew of contact between Pólya and Escher but the correspondence had been lost. She had interviewed Pólya before his death in 1985 and he recalled that there had been letters, but he couldn't find them. People assumed that if he had received letters from Escher they had been lost in his move from Zürich to America in 1940. The subsequent part of this story is part of the reason for an Afterword in this new edition.

Scholars assumed that Escher, who was largely unschooled in mathematics, had been influenced by Pólya's 1924 paper, but evidence and the details of the influence were not readily at hand. Roughly ten years after Pólya's death the new owner of Pólya's Palo Alto house, following the death of Pólya's widow, discovered two suitcases and a large box of papers and reprints in the attic, all sadly somewhat affected by age, neglect and the damp. The material then disappeared for a few years only to reappear in 2002. Upon examination this reviewer discovered that much of it dated from the 1930s when the correspondence between Pólya and Escher was supposed to have taken place.

The excitement was short-lived, however. The letters did not show up, but in some ways, what was there was even more interesting. Schattschneider discovered drawings by Pólya showing repeating patterns of snakes eating snakes eating snakes... The pattern was relatively simple compared to Escher's more sophisticated patterns. But on the back of one drawing was a notation in Pólya's hand indicating that he had sent it to Escher at a Brussels address. Beyond that, however, there were notes and sketches, with the outline of a table of contents, indicating serious work by Pólya on the manuscript of a book on symmetry for the lay person. There were sketches of tiles, probably made on trips to the Alhambra and elsewhere in Europe where he was looking at tiles and other examples of symmetry. Had Pólya completed his book and published it, he would have anticipated by fifteen years or so Hermann Weyl's classic Symmetry, a book also aimed at a general audience. The project was probably abandoned in the rush to get to America and get away from what was perceived at that time as a threat by the Nazis even to Switzerland, and the difficulties once here of finding an academic position.

Schattschneider chronicles the discovery of these papers in her new Afterword for this edition. Her tone is straightforward and factual — one regrets only that she does not quite convey the excitement of the sleuthing it took to track down these new pieces of information. This is only one of the stories of new developments in the saga of Escher's evolving ideas in constructing his amazing tilings. There are accounts of many problems now solved, even the use of the computer in producing Escher-like patterns. Schattschneider raises the question of whether Escher's work would have been very different had Escher had available to him modern computers and software. She asserts that in her view it would not be much different: he "chose simple time-tested tools, relying on hand and eye to realize his ideas."

While Schattschneider was writing this new material for the Afterword, others were discovering new Escher-like things: Marjorie Rice (of whom Schattschneider has written before), Makoto Nakamura, and Rinus Roelofs. The Escher legacy lives on. Had Schattschneider waited just a bit longer she might have included some recent work of H. W. Lenstra, Jr. (Notices Amer. Math. Soc. 50 (4) (2003), 446-455). But, then, something new needs to go into the next edition of Schattschneider's book.

The concordance, bibliography, various indexes, and the owners lists have all been updated. It's an impressive piece of scholarship that is extraordinarily beautiful as well. This book is an old friend and it's good to welcome it back in such an elegant and sumptuous form.

Gerald L. Alexanderson is Valeriote Professor of Science at Santa Clara University. He has served as Editor of Mathematics Magazine and as Secretary and President of the MAA.

The table of contents is not available.