Mathematics and the Art of M.C. Escher

When selecting courses for a specific major, students will notice that classes usually gravitate to either art or mathematics and science. Over time intellects have separated the creative ideas of the artists from the logical and structured ideas of the scientists. Rarely does one find “Introduction to color and design” as a requirement for a computer programming major. Similarly, society itself has had a tendency to take a similar attitude of art and mathematics. However, what if an artist coupled his creative ideas with that of mathematic principle? In that case, the artist would be M.C. Escher.

Maurits Cornelius Escher, born in 1898 in Leeuwarden, Holland, (The Mathematical Art of M.C. Escher) was a book illustrator and a designer for tapestries, postage stamps, and murals. However, Escher is most notably remembered for being a graphic artist. (M.C. Escher by Cordon Art b.v.). Throughout his lifetime, Escher has created fascinating works of art through the use of mathematical ideas (The Mathematical Art of M.C. Escher).

As a child, M.C. Escher studied to become an architect like his father. However, because of his drawing ability, he was attracted to the field of graphic art. Sadly, as with many artists, M.C. Escher’s art went unnoticed at first. However, in 1956, he had his first important exhibition, drawing even the attention of Time magazine. Because of this exhibit and review, M.C. Escher captured worldwide attention. Most importantly, Escher gained the attention of mathematicians who understood his extensive use of mathematic principles.

Encouraged by this recognition, M.C. Escher continued to study mathematical ideas and to use concepts such as structures in plane and projective geometry, as well as the use of non-Euclidean geometries. Moreover, fascinated with the ideas of Roger Penrose, Escher experimented with paradox and “impossible” figures. It is no surprise then that M.C. Escher decided to deal with the areas of geometry of space in addition to the logic of space.

One major characteristic of M.C. Escher’s art is his extensive use of geometric tessellations. Tessellations are “arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps” (The Mathematical Art of M.C. Escher). Basic geometric shapes are common in his work, especially prism-like forms (Abrams 7). Contrary, M.C. Escher preferred to use irregular shapes that changed and interacted to create, in his words, “metamorphoses” (The Mathematical Art of M.C. Escher). Escher’s interest in mathematical ideas and the use of tessellations began in 1936 when he studied the tile patterns in the Alhambra during his visit to Spain. According to Escher, this “was the richest source of inspiration that I have ever tapped” (The Mathematical Art of M.C. Escher). It was at this time, he noticed that traditional tessellations only used triangles, squares, and hexagons as patterns. Using his unique perspective, M.C. Escher filled these shapes with unusual objects. In his simpler designs he placed simple objects inside the repeating triangle pattern. However, in his more complicated works, he would create a changing scene through use of the hexagonal pattern, using the geometric principles of reflections, glide reflections, translations, and rotations. Moreover, he distorted his basic objects, such as birds and reptiles to change them into new figures. Even through this interesting method of distortion, he required that they obey the three, four or six-fold symmetry (The Mathematical Art of M.C. Escher).

Another aspect of geometry in his works, is M.C. Escher’s use of polyhedra. In mathematical terms, there are only five polyhedra with exactly similar polygonal faces. These Platonic Solids are: 1) the tetrahedron, which has four triangle faces; 2) the cube, which has six square faces; 3) the octahedron, which has eight triangle faces; 4) the dodecahedron, which has twelve pentagonal faces; and 5) the icosahedron, which has twenty-eight triangle faces. Remarkably, M.C. Escher was able to use four polyhedra in his piece Four Regular Solids. Furthermore, he intersected these transparent shapes so that their symmetry aligned. Similarly, Escher was also able to stellate a solid, or replace each of the faces with a pyramid (The Mathematical Art of M.C. Escher). In his work, Order and Chaos, he made a stellated dodecahedron within the crystalline sphere (The Mathematical Art of M.C. Escher).

Escher handled not only areas of geometry of space in his art, but also logic of space. It has been stated that this concept is among the most important of his art, mathematically speaking. Three Intersecting Planes is a good example of M.C. Escher’s concerns with the dimension of space, discerning three-dimensional concepts in a two-dimensional representation. Furthermore, in Mobius Strip II, Escher explores the aspects of Topology, which deals with “properties of space which are unchanged by distortion which may stretch or bend it – but which do not tear or puncture it” (The Mathematical Art of M.C. Escher).

Another way that Escher used logic of space was by creating optical illusions, or visual paradoxes. Through his studies, Escher learned that “the geometry of space determines its logic, and likewise the logic of space often determines its geometry” (The Mathematical Art of M.C. Escher). Therefore, through his use of light and shadow, as seen in Cube with Ribbons, as well as his use of perspective, as seen in High and Low, M.C. Escher has certainly created visual paradoxes that leave the viewer to marvel.

Considering that art and mathematics are separated into opposite categories makes Escher’s creativity even more astonishing. His art, intertwining such intense mathematical concepts, leaves many pondering whether he should be classified as an artist or a mathematician. Perhaps that should be left for each individual to decide, for even M.C. Escher said, “For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art” (The Mathematical Art of M.C. Escher)

Abrams, Harry N. The World of M.C. Escher. New York, NY: Harry N. Abrams, Inc.,

Publishers, 1971.

“The Mathematical Art of M.C. Escher.” Math Academy Online, 1997-2001.

< http://www.mathacademy.com/pr/minitext/escher>

“M.C. Escher by Cordon Art b.v.” Cordon Art B.V. < http://www.mcescher.com>