M. C. Escher: Master of Tessellation

M.C. ESCHER: Art and Science. H.S.M. Coxeter, M. Emmer, R. Penrose and M.L Teuber, eds. Elsevier, New York, 1986. 402 pp. $50. Who has failed to notice that exposition in the mathematical sciences is more pictorial lately? Today, it is not uncommon to find technical mathematics illustrated with drawings. And not just with elaborate symbols which, before the morning coffee, resemble an intimidating jumble. There have even appeared in august mathematical journals whole articles consisting entirely of pictures—the conventional, written exposition would require many more pages and make for duller reading. The pictures often tell all. Let there be no mistake about it. Many of the pictures that now routinely appear in print are no more than pictorial aids to reasoning, graphical sketches intended to suggest or persuade rather than convince. This is a far cry from the prevailing orthodoxy of only a few decades ago. It has always been a widespread practice, enjoyed by most working mathematicians in the privacy of the study, to use pictorial images of one kind or another to aid in understanding and to prompt discovery. The very same pictures rarely appeared in the final polished manuscript. The canons of scientific writing required formal exposition, often obscuring what was, upon a time, simple and clear. Indeed, although pictorial reasoning in mathematics may have prevailed up to the middle of the 19th century, it fell from favor by the beginning of the 20th century, when lessons then learned showed that pictures might mislead. Even in geometry itself pictorial reasoning has been called into question. For 2,000 years no one has succeeded in proving that Euclid's fifth postulate was a logical consequence of the first four yet, until the work of Bolyai and Lobatchevsky, in the 1820s, a simple sketch was generally sufficient to convince even the skeptical. A time-honored approach to mathematical argument—using examples and counterexamples, appealing to intuition and common sense, invoking mental pictures—suddenly seemed filled with pitfalls. Why are pictures now commonplace in mathematical exposition? Computer science with its heavy reliance on discrete mathematics and combinatorics is an important reason. Combinatorics thrives on pictures and pictorial reasoning. And computers are, of course, much on everyone's mind today. There is an artist whose pictures non-mathematicians commonly associate with mathematical ideas—Maurits Cornelis Escher. He was a master of tessellation, or tiling of the plane—fitting together copies of a single figure so that a plane is covered without any gaps or overlap. A region of this plane would look rather like a jigsaw puzzle whose pieces are all identical. M.C. Escher: Art and Science is a remarkable book that will be of considerable interest to many in the mathematical sciences and beyond. It consists of nearly 40 articles (and many graphic illustrations) of interest to mathematicians, physicists, crystallographers, chemists, biologists, psychologists, psychiatrists, art historians and experts in computer graphics and visual communication. The articles are based on lectures delivered at an interdisciplinary congress held in Rome in 1985. Inevitably, many of the articles have a mathematical bent and indeed, some are written by the leaders of the mathematical study of tessellations. H.S.M. Coxeter writes on "Coloured Symmetry," B. Grunbaum on "Mathematical Challenges in Escher Geometry," G.C. Shephard on "What Escher Might Have Done," and R. Penrose on "Escher and the Visual Representation of Mathematical Ideas." The full set of articles is divided into several sections: Escher and Symmetry; Escher, Mathematics, and Visual Perception; Escher and Geometry; Escher, Cinema and Computer Graphics; Escher and the Physical World; Escher and Art; Escher and the Humanities. They are fairly accessible and generally well-written, and many are fascinating. Escher's work spans half a century, starting in the 1920s, yet it was not until late in his life that his work attracted the attention of the mathematical community in any serious way. M.C. Escher: Art and Science is itself a measure of the shift that has occurred in the mathematical sciences. Have the graphic arts suffered from their long separation from the mathematical sciences? We will, of course, never know; but one hopes their union today will not spoil them. For mathematics, though, the book is an occasion for cheer. It is time for the mathematical sciences to come out of the drawing room with polished pictures firmly in hand for ready communication. Rival is a professor of pure mathematics and computer science at the University of Ottawa, Ontario, Canada KiN 9B4. He is the editor of Order: A Journal on the Theory of Ordered Sets and Its Applications.

M. C. Escher: Master of Tessellation

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