Maurits Cornelis Escher: His Influence in Math and Art

Maurits Cornelis Escher was born in Leeuwarden, Holland on June 17th, 1898. He was a master artist and creative visionary. He was fascinated by a wide range of mathematical concepts and incorporated them directly into his work. Escher himself affirmed that, "Although I am absolute innocent of training or knowledge in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists." Although Escher had no formal training in mathematics, his work displayed an innate understanding of mathematical principles including the geometry of space and the logic of space. He contributions to both the realms of mathematics and art were significant.

In 1936, Escher traveled to Granada, Spain and became intrigued by the Moorish mosaics in the Alhambra. His interest in these mosaics inspired his studies in transformational geometry, focusing on the regular divisions of the plane, also known as "tessellations." Tessellations are patterns of closed shapes that completely cover the plane without overlapping and without leaving gaps. While mathematicians had shown that only the triangle, square, and hexagon could be used for tessellations, Escher "exploited these basic patterns in his tessellations, applying what geometers would call reflections, glide reflections, translations, and rotations to obtain a greater variety of patterns" (Platonic Realms 2007). In addition, his extensive research in the geometry of space was years ahead of its time. He unknowingly contributed to the understanding and development of crystallography by creating his own categorization system covering all possible combinations of shape, color, and symmetrical properties of color-based division; this information was published in his now publicly-recognized paper "Regular Division of the Plane with Asymmetric Congruent Polygons" in 1941.

By 1956, Escher's interest in the regular division of the plane expanded as he explored the nature of space itself, capturing the notion of hyperbolic space (the concept of infinity) on a fixed 2-dimensional plane. With the help of the Canadian mathematician H.S.M Coxeter and Poincare's circle model, Escher was able to accurately represent hyperbolic tessellations in his latest work. Because of the hard work and dedication required to complete these projects, Escher found himself incredibly satisfied with his progress, noting that "I discovered once again that the human hand is capable of executing small and yet completely controlled movements, on the condition that the eye sees sufficiently clearly what the hand is doing" (Strauss 1996). Coxeter himself was impressed by the preciseness of Escher's work, calling attention to Circle Limits III in a paper he published in 1995: "... [Escher] got it absolutely right to the millimetre, absolutely to the millimetre .... Unfortunately he didn't live long enough to see my mathematical vindication" (Schattschneider 91). In addition, Escher's Circle Limits prints have been used by many mathematicians as instruments for strengthening understanding and teaching hyperbolic geometry.

Along with his studies in the geometry of space, Escher explored the logic of space. This can be explained as the "spatial relations among physical objects which are necessary, and which when violated result in visual paradoxes" (Platonic Realms 2007). These visual paradoxes are also referred to as "optical illusions," and consist of manipulating light and shadow across concave and convex objects in order to achieve misleading and deceptive visuals. With the assistance of the British mathematician Roger Penrose, Escher gained a keen understanding of the relationship between the logic of space and the geometry of space, and vividly represented it in many of his works such as Waterfall and Up and Down. According to Eric Weisstein, Escher himself was the inspiration for the very creation of the Penrose triangle, also called the tribar (2007).

The importance of Escher's contributions to mathematics is immeasurable. In addition to active research and collaboration with various mathematicians such as Roger Penrose, J. F. Schouten, H S. M. Coxeter, and J. W. Wagenaar in the fields of topology, optical illusions, and hyperbolic tessellations, Escher was also ahead of several other mathematicians in various fields. For example, research conducted by Russian crystallographers in 1951 regarding polychromatic symmetry had already been thoroughly (although unknowingly) investigated by Escher. In addition, his exploration of the "fundamental region" in his work with tessellations was years ahead of the research conducted by the German mathematicians Heinrich Heesch and Otto Kienzle. Escher's work has also been used for many instructional purposes: in 1960 x-ray crystallographers speaking at the Fifth International Congress of the International Union of Crystallography presented some of Escher's work in order to explain symmetry and transformations. In 1961, The Nobel Prize winner Chen Ning Yang used Horsemen to explain the symmetry operations in particle physics (Escher, Emmer, and Schattschneider 114).

Escher has also made significant contributions to the future of art. According to the artist and scientist Wayne Roberts, "[Escher] almost single-handedly united geometry, number and art so logically that he fitted them together literally as pieces of a jig-saw puzzle" (par. 2). Douglas R. Hofstadter also noted that, "there is much more to a typical Escher drawing than just symmetry or pattern; there is often an underlying idea, realized in artistic form" (11). He was not afraid to explore art in its purest form. Through his work he demonstrated that creativity was limitless-- the only boundaries that existed were the ones that the artist constructed himself.

Escher died on March 27th, 1972 in Laren, Netherlands. During his lifetime he created 448 lithographs, woodcuts and wood engravings, and over 2000 drawings and sketches. He was considered to be the creator "of some of the most intellectually stimulating drawings of all time" (Hofstadter 10). But he contributed far more than that: through his exploration of the logic and geometry of space he was able to successfully interconnect both the structure and syntax of visual form, thereby opening up new fields of research, expanding current theories, and bringing forth an entirely new perspective on mathematics.

References

Douglas R. Hoftstadter. "Godel, Escher, Bach: An Eternal Golden Braid." Vintage Books, 1980. p. 10-11.

Maurits Cornelis Escher, Michele Emmer, and Doris Schattschneider. "M.C. Escher's Legacy: A Centennial Celebration". Springer, 2003. p. 114.

Weisstein, Eric W. "Penrose Triangle." From MathWorld--A Wolfram Web Resource. 21 Oct. 2007.
http://mathworld.wolfram.com/PenroseTriangle.html>.

Platonic Realms. "Mathematical Art of M.C. Escher." 17 Oct. 2007.
http://www.mathacademy.com/pr/minitext/escher/>.

Totally Tessellated. "Escher Biography." 17 Oct. 2007.
http://library.thinkquest.org/16661/escher/biography.3.html>.

S Strauss, M C Escher. The Globe and Mail. 9 May 1996.

D Schattschneider, "Escher: A mathematician in spite of himself, in R K Guy and R E Woodrow (eds), The Lighter Side of Mathematics". Washington, 1994. p. 91-100.