Through the Garden Gate Opened by Crystallographers

December 26, 2004

Figure 1. George Plya's illustration representing the 17 plane symmetry groups, from his article "ber die Analogie der Kristallsymmetrie in der Ebene," which appeared in Zeitschrift fr Kristallographie in 1924. Four of the designs---C1, C3, C4, and D1gg---are Plya's original creations. From M.C. Escher: Visions of Symmetry.

Book Review
James Case

M.C. Escher: Visions of Symmetry. By Doris Schattschneider, Harry N. Abrams, New York, 2004, 370 pages, $30.00.

When was the last time you purchased an attractive full-color coffee-table art book for less than thirty dollars? How much mathematical content did it include? If your answers are "not lately" and "not much," this revised and expanded edition of Doris Schattschneider's 1990 minor classic on the art of M.C. Escher may be the book for you. Covering only the artist's work on what he called "regular divisions of the plane," it reproduces many of his most arresting works, along with a large number of previously unpublished drawings from his notebooks.

The dustcover reveals that Doris Schattschneider received a PhD in mathematics from Yale and taught at Moravian College in Bethlehem, Pennsylvania, for thirty-four years. She is the author of numerous mathematical papers, a former editor of Mathematics Magazine (a journal of undergraduate research published by the Mathematical Association of America), and the 1993 recipient of the national MAA Award for Distinguished Teaching of College or University Mathematics. She has also lectured globally on a variety of geometric subjects. She first viewed Escher's collected work on the regular division of the plane in 1976, and resolved---more or less on the spot---to tell the story behind it.

At the time, Escher's notebooks, sketchbooks, and folio drawings were still owned by the Escher Foundation and held for safekeeping at a Dutch museum. In 1980, the Escher Foundation was dissolved, and its assets were sold to qualified bidders the world over. The work on regular divisions of the plane now resides on four continents, more than half of it in private hands. Schattschneider declares, in her preface to the original edition, "This book attempts to reunite that body of work and tell its story from Escher's point of view." The new edition includes an expanded bibliography and an afterword describing recent developments. Among these activities are several 1998 exhibitions and conferences celebrating the centennial of Escher's birth; the construction of a new museum devoted to his work; the dedication of several thousand Internet sites; and a body of work by scientists, mathematicians, artists, and teachers who have pursued his ideas in a variety of ways. (See Sara Robinson's article "M.C. Escher: More Mathematics Than Meets the Eye," SIAM News, October 2002.

Maurits Cornelis Escher was born in 1898, at Leeuwarden, the Netherlands. The youngest of his father's five sons, and his mother's three, he received most of his schooling at Arnhem, to which the family moved soon after his birth. His father and three of his brothers were trained in what he once described in a letter as "the exact sciences or engineering," and he retained a lifelong respect for their quantitative approach to the world around them. In the fall of 1919, after spending time at the Higher Technical School in Delft, Escher enrolled at the School for Architecture and Decorative Arts in Haarlem, intending to prepare for a career in architecture. Once there, however, he was soon persuaded to specialize in the graphic arts.

Escher remained at Haarlem until the spring of 1922, when he left for Italy. There he met and married fellow artist Jetta Umiker; the couple had three sons. They settled in Rome and remained there until July 1935, when the vicissitudes of life under the fascisti caused them to decamp for the Swiss mountain village of Chateau d'Oex. From there they moved to Ukkel, a suburb of Brussels, which they soon left to settle more or less permanently in the peaceful Dutch village of Baarn.

From his base in Italy, Escher had traveled extensively. In Spain, he encountered the colorful geometric patterns with which the 11th- and 12th-century Moors had chosen to decorate their castles and public buildings. On what they considered a "farewell to the Mediterranean" trip in 1936, he and his wife copied, in pencil, colored pencil, watercolor, and ink, as many as they could of the Moorish designs in the Alhambra (at Granada), La Mexquita (Cordoba), and the (less appealing) Alcazar (Seville). Escher's efforts to create similar designs of his own were soon to redirect his artistic career onto an entirely new path.
In October 1937, during a visit to his parents' home in the Hague, he chanced to show some of his work involving regular divisions of the plane to his half brother B.G. (Beer) Escher, a professor of geology at the University of Leiden. Immediately seeing the connection with two-dimensional crystallography, Beer advised his sibling to consult Zeitschrift fr Kristallographie to learn more. A week later Beer provided a list of ten papers on the subject, all written in German and published in that journal between 1911 and 1933. Most were too theoretical for a layman, but one proved to contain exactly what Escher was looking for. Published in 1924 by George Plya, the paper showed that repetitive designs in the plane can be classified by symmetry group---of which there are just 17---and included a page (see Figure 1) illustrating all possibilities. It is a pity that Beer did not direct his half brother to Hilbert and Cohn-Vossen's popularization Geometry and the Imagination, which would also have given him the information he required.

From a paper by F. Haag, Escher learned that theorems asserting the existence of only a few tilings of the plane meant tilings by convex subsets only. Without that restriction, it would be possible to produce all manner of variations on each individual mathematical theme. Escher's notebooks soon began to reflect this realization, as shown in Figure 2. In time, Escher's interests came to include regular colored tilings, as well as tilings of the hyperbolic plane. His investigations of the former anticipated the work of crystallographers by at least twenty years and brought him numerous invitations to address scientific audiences.

Figure 2. Escher's notebooks reflect his exploration of the crystallographic literature.

"Crystallographers," Escher once asserted, "have . . . ascertained which and how many ways there are of dividing a plane in a regular manner. In doing so, they have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature, they are more interested in the way the gate is opened than in the garden that lies behind it." Escher spent almost half his working life exploring and illustrating that garden.

In his writings, as well as his public lectures, he often wondered aloud whether (and/or why) others had not anticipated his concern with regular divisions of the plane. "If it counts as art," he once inquired, "why has no other artist---as far as I have been able to discover---ever occupied himself deeply with it? Why am I the only one captivated by it?" Over and over, in his books, articles, papers, and public lectures, Escher reported having searched high and low for artistic examples of regular divisions of the plane, with but little success. Could none of his listeners help him to locate more? Schattschneider reports that he had several precursors in the Art Nouveau movement of the late 19th century; she includes two examples---both by a relatively prominent Viennese artist named Koloman Moser---and wonders why Escher did not learn of them during his years at the Haarlem school.

Almost anyone viewing Escher's art for the first time, especially anyone with mathematical training, has to wonder how he managed to achieve so many novel visual effects. Schattschneider provides the most credible and complete answers we are ever likely to get for questions of that sort. Anyone else who has entertained such questions will find her answers interesting and---at a guess---largely satisfactory. The information she has gathered, together with 440 striking illustrations, makes a unique and appealing package.

James Case writes from Baltimore, Maryland.

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