Escher's Metaphors The prints and drawings of M.C. Escher give expression to abstract concepts of mathematics and science by Doris Schattschneider Throughout his life Maurits Cornelis Escher (he used only M.C.) remarked on his inability to understand mathematics, declaring himself ‘absolutely innocent of training or knowledge in the exact sciences'. Yet even as a child, Escher was intrigued by order and symmetry. The fascination later led him to study patterns of tiles at the Alhambra in Granada, to look al geometric drawings in mathematical papers (with the advice of his geologist brother) and ultimately to pursue his own unique ideas for tiling a plane. Escher's attention to the coloring of his drawings of interlocked tiles anticipated the later work of mathematicians and crystallographers in the field of color symmetry. His works are now commonly used to illustrate these concepts. His exhibit in conjunction with the 1954 International Congress of Mathematicians in Amsterdam and the publication of his first book (The Graphic Work of M.C. Escher) in 1959 struck a chord with mathematicians and scientists that still resonates strongly. He wrote that a main impetus for his work was ‘a keen interest in geometric laws contained by nature around us'. In expressing his ideas in graphic works, he provided arresting visual metaphors for fundamental ideas in science. Escher was born in 1898 in the town of Leeuwarden, Holland. The youngest son of a civil engineer, he grew up with four brothers in Arnhem. Although three of his brothers pursued science or engineering, Escher was a poor mathematics student. With the encouragement of his high school art teacher, he became interested in graphic arts, firts making linoleum cuts. In 1919 he entered the School for Architecture and Decorative Arts in Haarlem, intending to study architecture. But when he showed his work to Samuel Jessurun de Mesquita, who taught graphic arts there, he was invited to concentrate in that field. De Mesquita had a profound influence on Escher, both as a teacher (particularly of woodcut techniques) and later as a friend and fellow artist. After finishing his studies in Haarlem, Escher settled in Rome and made many extensive sketching tours, mostly in southern Italy. His eyes discerned striking visual effects of the ordinary - architectual details of monumental buildings from unusual vantage points, light and shadow cast by warrens of staircases in tiny villages, clusters of houses clinging to mountain slopes that plunged to distant valleys and, at the opposite scale, tiny details of nature as if viewed through a magnifying glass. In his studio, he would transform the sketches into woodcuts and lithographs. In 1935 the political situation became unendurable, and with his wife and young sons, Escher left Italy forever. After two years in Switzerland and then three years in Uccle, near Brussels, they settled permanently in Baarn, Holland. These years also brought an abrupt turn in Escher's work. Almost all of it from this time would draw its inspiration not from what his eyes observed but rather from his mind's eye. He sought to give visual expression to concepts and to portray the ambiguities of human observation and understanding. In doing so, he often found himself in a world governed by mathematics. Escher was fascinated, almost obsessed, with the concept of the ‘regular division of the plane'. In his lifetime, he produced more than 150 color drawings that testified to his ingenuity in creating figures that crawled, swam and soared, yet filled the plane with their clones. These drawings illustrated symmetries of many different kinds. But for Escher, division of the plane was also a means of capturing infinity. Although a tiling such as the one using butterflies can in principle be continued indefinitely, thus giving a suggestion of infinity, Escher was challenged to contain infinity within the confines of a single page. "Anyone who plunges into infinity, in both time and space, farther and farther without stopping, needs fixed points, mileposts as he flashes by, for otherwise his movement is indistinguishable from standing still," Escher wrote. "He must mark off his universe into units of a certain length, into compartments which repeat one another in endless succession." After completing several prints in which figures endlessly diminish in size as they approach a cetnral vanishing point, Escher sought a device to portray progressive reduction in the opposite direction. He wanted figures that repeated forever, always approaching - yet never reaching - an encircling boundary. In 1957 the mathematician H.S.M. Coxeter sent Escher a reprint of a journal article in which he illustrated planar symmetry with some of Escher's drawings. There Escher found a figure that gave him ‘quite a shock' - a hyperbolic tesselation of triangles that showed exactly the effect he sought. From careful study of the diagram, Escher discerned the rules of tiling in which circular arcs meet the edge of an encompassing circle at right angles. During the next three years, he produced four different prints based on this type of grid, of which Circle Limit IV was the last. Four years later Escher devised his own solution to the problem of infinity within a rectangle. His recursive algorithm - a set of directions repeatedly applied to an object - results in a self-similar pattern in which each element is related to another by a change of scale. Escher sent Coxeter a sketch of the underlying grid, apologizing:"I fear that the subject won't be very interesting, seen from your mathematical point of view, because it's really simple as a flat filling. Nonetheless it was a headaching job to find an adequate method to realise the subject in the simplest possible way". In a lecture a few summers ago mathematician William P. Thurston, director of the Mathematical Sciences Research Institute at the University of California at Berkeley, illustrated the concept of self-similar tiling with just such a grid, unaware of Escher's earlier discovery. Curiously, self similar patterns provide examples of figures that have fractional, or fractal, dimension, an ambiguity that Escher would doubtlessly have enjoyed. In 1965 he confessed:"I cannot help mocking all our unwavering certainties. It is, for example, great fun delibarately to confuse two and three dimensions, the plane and space, and to poke fun at gravity". Escher was masterful at confusing dimensions, as in Day and Night, in which two-dimensional farm fields mysteriously metamorphose into three-dimensional geese. He also delighted in pointing out the ambiguities and contradictions inherent in a common practice of science: pasting together several local views of an object to form a global whole. Near the end of his life (he died in 1972), Escher wrote,"Above all, I am happy about the contact and friendship of mathematicians that resulted from it all. They have often given me new ideas, and sometimes there is even an interaction between us. How playful they can be, these learned ladies and gentlemen!" From: Scientific American, November 1994 Further reading: Coxeter, H.S.M. (1981). Angels and devils. Klarner, D.A. (Ed.)The Mathematical Gardner. Weber and Schmidt. Coxeter, H.S.M., Emmer, M., Penrose, R. & Teuber, M.L. (Eds.) (1986). M.C. Escher: Art and Science. North Holland. Ernst, B. (1976). The magic mirror of M.C. Escher. Random House. Escher, M.C. (1971). The graphic work of M.C. Escher. Ballantine Books. Escher, M.C. (1989). Escher on Escher: exploring the infinite. Translated by Karin Ford. Harry N. Abrams. Schattschneider, D. (1990). Work of M.C. Escher. W.H. Freeman and Company. Locher, J.L. (Ed.) (1989). M.C. Escher: his life and complete graphic work. Harry N. Abrams. back