## Mathematical art of M.C. Escher |

## Introduction
Maurits Cornelis Escher, who was born in Leeuwarden, Holland in 1898, created unique and fascinating works of art that explore and exhibit a wide range of mathematical ideas. While he was still in school his family planned for him to follow his father's career of architecture, but poor grades and an aptitude for drawing and design eventually led him to a career in the graphic arts. His work went almost unnoticed until the 1950’s, but by 1956 he had given his first important exhibition, was written up in Time magazine, and acquired a world-wide reputation. Among his greatest admirers were mathematicians, who recognized in his work an extraordinary visualization of mathematical principles. This was the more remarkable in that Escher had no formal mathematics training beyond secondary school.
As his work developed, he drew
great inspiration from the mathematical ideas he read about, often working
directly from structures in plane and projective geometry, and eventually
capturing the essence of non-Euclidean geometries, as we will see below. He was
also fascinated with paradox and "impossible" figures, and used an
idea of Roger Penrose’s to develop many intriguing works of art. Thus, for the
student of mathematics, Escher’s work encompasses two broad areas: the
geometry of space, and what we may call the ## Tesselations
Regular divisions of the plane, called "tessellations", are arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps. Typically, the shapes making up a tessellation are polygons or similar regular shapes, such as the square tiles often used on floors. Escher, however, was fascinated by every kind of tessellation – regular and irregular – and took special delight in what he called "metamorphoses," in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself. His interest began in 1936, when he traveled to Spain and viewed the tile patterns used in the Alhambra. He spent many days sketching these tilings, and later claimed that this "was the richest source of inspiration that I have ever tapped." In 1957 he wrote an essay on tessellations, in which he remarked:
Whether or not this is fair to the mathematicians, it is true that they had
shown that of all the regular polygons, only the triangle, square, and hexagon
can be used for a tessellation. (Many more
In ## Polyhedra
The regular solids, known as
There are many interesting solids that may be obtained from the Platonic solids
by intersecting them or stellating them. To
Intersecting solids are also represented in many of Escher's works, one of the
most interesting being the wood engraving Here are solids constructed of intersecting octahedra, tetrahedra, and cubes,
among many others. One might pause to consider, that if Escher had simply drawn
a bunch of mathematical shapes and left it at that, we probably would never have
heard of him or of his work. Instead, by such devices as placing the chameleons
inside the polyhedron to mock and alarm us, Escher jars us out of our
comfortable perceptual habits and challenges us to look with fresh eyes upon the
things he has wrought. Surely this is another source of the mathematicians'
admiration for Escher's work – for just such a perceptual ## The shape of space
Among the most important of Escher's works from a
mathematical point of view are those dealing with the nature of space itself.
His woodcut
Inspired by a drawing in a book by the mathematician H.S.M Coxeter, Escher
created many beautiful representations of hyperbolic space, as in the woodcut
Even more unusual is the space suggested by the woodcut In addition to Euclidean and
non-Euclidean geometries, Escher was very interested in visual aspects of
Topology, a branch of mathematics just coming into fullflower during his
lifetime. Topology concerns itself with those properties of a space which are
unchanged by distortions which may stretch or bend it – but which do not tear
or puncture it – and topologists were busy showing the world many strange
objects. The Möbius strip is perhaps the prime example, and Escher made
many representations of it. It has the curious property that it has only one
side, and one edge. Thus, if you trace the path of the ants in
Another very remarkable lithograph, called All of Escher's works reward a prolonged stare, but this one does especially.
Somehow, Escher has turned space back into itself, so that the young man is both
inside the picture and outside of it simultaneously. The secret of its making
can be rendered somewhat less obscure by examining the grid-paper sketch the
artist made in preparation for this lithograph. Note how the scale of the grid
grows continuously in a clockwise direction. And note especially what this trick
entails: A hole in the middle. A mathematician would call this a ## The logic of space
By the "logic" of space we mean those spatial
relations among physical objects which are
Escher understood that the geometry of space determines its logic, and likewise
the logic of space often determines its geometry. One of the features of the
logic of space which he often applied is the play of light and shadow on concave
and convex objects. In the lithograph Another of Escher's chief concerns was with perspective. In any perspective drawing, vanishing points are chosen which represent for the eye the point(s) at inifinity. It was the study of perspective and "points at infinity" by Alberti, Desargues, and others during the renaissance that led directly to the modern field of projective geometry. By introducing unusual vanishing points and forcing elements of a composition
to obey them, Escher was able to render scenes in which the "up/down" and
"left/right" orientations of its elements shift, depending on how the
viewer’s eye takes it in. In his perspective study for
A third type of "impossible drawing" relies on the brain's insistence upon using visual clues to construct a three-dimensional object from a two-dimensional representation, and Escher created many works which address this type of anomaly. One of the most intriguing is based on an idea of the mathematician Roger
Penrose’s – the impossible triangle. In this lithograph, ## Self-reference and information
Our final consideration of Escher's art involves its
relationship to the fields of information science and artificial intelligence.
This aspect of his work has been largely overlooked in previous studies, but the
case for its importance to these fields was forcefully made by Douglas R.
Hofstadter in his 1980 Pulitzer Prize winning book,
A central concept which Escher captured is that of self-reference, which many believe lies near the heart of the enigma of consciousness – and the brain's ability to process information in a way that no computer has yet mimicked successfully. The lithograph
On a deeper level, self-reference is found in the way our worlds of perception
reflect and intersect one another. We are each like a character in a book who is
reading his or her own story, or like a picture of a mirror reflecting its own
landscape. Many of Escher's works exhibit this theme of intersecting worlds, but
we will here consider only one of the exemplars. As is common in Escher's
treatment of this idea, the lithograph And so we end where we began, with a self portrait: the work a reflection of the artist, the artist reflected in his work. ## ConclusionWe have here considered only a handful among the hundreds of drawings, lithographs, woodcuts, and mezzotints Escher left to us upon his death in 1972. Much more could be said, and has been said, about the depth, meaning, and importance of his work. The reader is encouraged to explore further the rich legacy of M.C. Escher, and to ponder anew the intersections he has drawn for us among the world of imagination, the world of mathematics, and the world of our waking life. Reprinted from http://www.mathacademy.com/pr/minitext/escher/ |