Escher+Project

M.C. Escher: A MATHEMATICAL ARTIST
by Natalie Gerardi

//By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I have made, I ended up in the domain of mathematics, Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. -M.C. Escher//

Have you ever seen a picture that stretches to infinity? Have you ever seen a picture that has rows and rows of the same shape? Have you ever seen a picture that sends your eyes and brain whirring with confusion? If the answer is yes, chances are you've seen a work by M.C. Escher, a well-known artist and mathematician. Maurits Cornelius Escher was born on June 18, 1898 in Leeuwarden, Holland. His family expected him to follow in his father's footsteps and become an architect, but Escher received poor grades in school, so this was not possible. Instead, his fascination with drawing led him to be a graphic artist. In his early years, Escher informally sketched landscapes and insects; in 1922 he produced his first artistic work, featuring eight human heads in different planes. By 1956 he had given his first important exhibition, and his popularity skyrocketed from there. At his death in 1972, at the age of 79, Escher had completed hundreds of drawings, lithographs, woodcuts, and mezzotints. And added to that, he had forever influenced and furthered the world of art, the world of science, and the world of mathematics. Is it possible that one man did all of this? How could he have used math in art? Did he even plan to contribute so much to mathematics? Using these and many more questions, I tried to find out how mathematics is reflected in the art of M.C. Escher.

TESSELLATIONS, POLYHEDRA, and ESCHER
"It's impossible for art to reflect math," you may be thinking. "They are two completely different subjects." While this may seem to be the truth, mathematical patterns are often seen in Escher's drawings. One of these patterns is regular division of the plane, or as they are more commonly known, **tessellations**. Escher was fascinated with both regular and irregular tessellations. One of his main inspirations was the Alhambra in Spain, a building which features many beautiful tessellations using basic patterns. In his art, Escher exploited these patterns using reflections, glide-reflections, translations, and rotations. He also distorted basic shapes to create animal-themed tessellations. Another main subject in many of Escher's works, as well as a secondary element in many more, is regular solids, or **polyhedra**. The five "platonic solids" are the only five polyhedra with exactly similar polygonal faces: the tetrahedron (4 triangular faces), the cube (6 square faces), the octahedron (8 triangular faces), the dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces). These solids feature prominently in Escher's sketches and woodcuts, either by intersecting with each other (as in "Stars" and "Four Regular Solids") or by being stellated ("Order and Chaos").

ESCHER's ILLUSIONS
Not all of Escher's art was simple patterns. Often he manipulated the properties and logic of space, as well as perspective, to create "impossible" drawings and illusions. **Topology**, a branch of mathematics only just becoming recognized in his lifetime, affiliates itself with the properties of space that, while they may be stretched or bent, remain unchanged. An example of this is the Mobius strip, a curious object with only one side and one edge, of which Escher sketched many times. The play of light and shadow on concave and convex objects was another one of the features of logic of space that he applied to his work, as can be seen in "Cube with Ribbons". Unusual vanishing points in perspective drawings was yet another one of Escher's "tricks". In "High and Low", there are five different vanishing points (representing points of infinity), which results that in half of the picture you are looking up, and in the other half you are looking down--at the same scene!

PLANNING AND BALANCE
With all this amazing and clever incorporation of patterns and shapes in his art, it would make sense to call Escher a mathematical genuis. What is unbelievable is that he had no formal training beyond secondary school. Thus, it is probably safe to say that he started out not meaning to be so mathematical, but as his work developed, he drew heavy inspiration from the mathematical ideas he researched/read about. And that leads to another question---is there a balanced blend between art and math in his work, or is there more of one? Truthfully, this is a matter of opinion. Escher himself didn't quite know. He is quoted in an article as saying, "For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art." Personally, I believe that there is a balance: patterns and shapes play a huge part of course, but color and style do as well. What do you think?

Conclusion: ESCHER's EFFECT
M.C. Escher has left an enormous influence on the world today. Many artists use his unique ideas in their own art, even today. Through his art, he discovered/anticipated many concepts before professional mathematicians, scientists, and crystallographers, which have greatly furthered study in these fields. In 1998, an international exhibition was held in Rome to mark the centennial of his birth. Artists, scientists, engineers, mathematicians, psychologists, and teachers gathered together to celebrate his inspiring legacy. Thus, it is easy to see how mathematics and many other fields are reflected and influenced by the art of M.C. Escher.

Class Picture Presentation

Bibliography: Coxeter, H.S.M; Emmer, M.; Penrose, R.; Teuber, M.L. __M.C. Escher: Art and Science__. North-Holland, 1985 "The Mathematical Art of M.C. Escher" http://www.mathacademy.com/pr/minitext/escher, 2008 "Totally Tessellated: Escher Biography" http://library.thinkquest.org/16661/escher/biography.3.html, 2008 "M.C. Escher's legacy" http://en.wikipedia.org/wiki/M._C._Eschers_legacy, 2008