# Mathematical art of M.C. Escher

# Introduction

Self Portrait |

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 *logic* of space.

# Tesselations

Alhambra scetch |

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:

In mathematical quarters, the regular division of the plane has been considered theoretically. . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature thay are more interested in the way in which the gate is opened than in the garden lying behind it.

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 *irregular* polygons tile the
plane - in particular there are many tessellations using irregular pentagons.)
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. He also elaborated these patterns by
"distorting" the basic shapes to render them into animals, birds, and other
figures. These distortions had to obey the three, four, or six-fold symmetry of
the underlying pattern in order to preserve the tessellation. The effect can be
both startling and beautiful.

Regular division of the plane with the birds |
Reptiles |
Cycles |
Development 1 |

In *Reptiles* the tessellating creatures playfully escape from the
prison of two dimensions and go snorting about the destop, only to collapse back
into the pattern again. Escher used this reptile pattern in many hexagonal
tessellations. In *Development 1,* it is possible to trace the developing
distortions of the square tessellation that lead to the final pattern at the
center.

# Polyhedra

Four Regular Solids |

The regular solids, known as *polyhedra*, held a
special fascination for Escher. He made them the subject of many of his works
and included them as secondary elements in a great many more. There are only
five polyhedra with exactly similar polygonal faces, and they are called the
Platonic solids: the *tetrahedron,* with four triangular faces; the *cube*,
with six square faces; the *octahedron*, with eight triangular faces; the *dodecahedron,*
with twelve pentagonal faces; and the *icosahedron,* with twenty triangular
faces. In the woodcut *Four Regular Solids,* Escher has intersected all but
one of the Platonic solids in such a way that their symmetries are aligned, and
he has made them translucent so that each is discernable through the others.
Which one is missing?

Order and chaos |

There are many interesting solids that may be obtained from the Platonic solids
by intersecting them or stellating them. To *stellate* a solid means to
replace each of its faces with a pyramid, that is, with a pointed solid having
triangular faces; this transforms the polyhedron into a pointed,
three-dimensional star. A beautiful example of a stellated dodecahedron may be
found in Escher's *Order and Chaos.* Here the stellated figure rests within
a crystalline sphere, and the austere beauty of the construction contrasts with
the disordered flotsam of other items resting on the table. Notice that the
source of light for the composition may be guessed, for the bright window above
and to the left of the viewer is reflected in the sphere.

Stars |

Intersecting solids are also represented in many of Escher's works, one of the
most interesting being the wood engraving *Stars.*

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 *freshness*
lies at the back of all great mathematical discovery.

# The shape of space

Three intersecting planes |

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 *Three Intersecting Planes* is a good place to begin a review
of these works, for it exemplifies the artist's concern with the dimensionality
of space, and with the mind's ability to discern three-dimensionality in a
two-dimensional representation. As we will see in the next section, Escher often
exploited this latter feature to achieve astonishing visual effects.

Circle limit III |

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 *Circle
Limit III.* This is one of the two kinds of non-Euclidean space, and the
model represented in Escher's work is actually due to the French mathematician
Poincaré. To get a sense of what this space is like, imagine that you are
actually *in* the picture itself. As you walk from the center of the
picture towards its edge, you will shrink just as the fishes in the picture do,
so that to actually reach the edge you have to walk a distance that, to you,
seems infinite. Indeed, to you, being inside this hyperbolic space, it would not
be immediately obvious that anything was unusual about it - after all, you
have to walk an infinite distance to get to the edge of ordinary Euclidean space
too. However, if you were a careful observer you might begin to notice some odd
things, such as that all similar triangles were the same size, and that no
straight-sided figure you could draw would have four right angles - that is,
this space doesn't have any squares or rectangles. A strange place indeed!

Snakes |

Möbius Strip II |

Even more unusual is the space suggested by the woodcut *Snakes.* Here the
space heads off to infinity both towards the rim and towards the center of the
circle, as suggested by the shrinking, interlocking rings. If you occupied this
sort of a space, what would it be like?

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 *Möbius
Strip II,* you will discover that they are not walking on opposite sides of
the strip at all - they are all walking on the same side. It is easy to make a
Möbius strip; just cut a strip of paper with scissors, give it a half-twist,
and then glue or tape the ends. What do you predict will happen if you attempt
to cut such a strip in two, lengthwise?

Print gallery |

Another very remarkable lithograph, called *Print Gallery,* explores
both the logic and the topology of space. Here a young man in an art gallery is
looking at a print of a seaside town with a shop along the docks, and in the
shop is an art gallery, with a young man looking at a print of a seaside

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 *singularity*,
a place where the fabric of the space no longer holds together. There is just no
way to knit this bizarre space into a seamless whole, and Escher, rather than
try to obscure it in some way, has put his trademark initials smack in the
center of it.

# The logic of space

Cube with Ribbons |

By the "logic" of space we mean those spatial
relations among physical objects which are *necessary*, and which when
violated result in visual paradoxes, sometimes called optical illusions. All
artists are concerned with the logic of space, and many have explored its rules
quite deliberately. Picasso, for instance.

High and Low |

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 *Cube with Ribbons,* the bumps on the
bands are our visual clue to how they are intertwined with the cube. However, if
we are to believe our eyes, then we cannot believe the ribbons!

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 *High and Low,*
the artist has placed five vanishing points: top left and right, bottom left and
right, and center. The result is that in the bottom half of the composition the
viewer is looking up, but in the top half he or she is looking down. To
emphasize what he has accomplished, Escher has made the top and bottom halves
depictions of the same composition.

Waterfall |

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, *Waterfall,*
two Penrose triangles have been combined into one impossible figure. One sees
immediately one of the reasons the logic of space must preclude such a
construction: the waterfall is a closed system, yet it turns the mill wheel
continuously, like a perpetual motion machine, violating the law of conservation
of energy. (Notice the intersecting cubes and octahedrons on the towers.)

# Self-reference and information

Drawing Hands |

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, *Gödel, Escher, Bach:
An Eternal Golden Braid.*

Fish and Scales |

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 *Drawing Hands* and the woodcut *Fish and Scales*
each captures this idea in a different way. In the former the self-reference is
direct and conceptual; the hands draw themselves much the way that consciousness
considers and constructs itself, mysteriously, with both self and self-reference
inseparable and coequal. In *Fish and Scales,* on the other hand, the
self-reference is more functional; one might rather call it self-resemblence. In
this way the woodcut describes not only fish but all organisms, for although we
are not built, at least physically, from small copies of ourselves, in an
information-theoretic sense we are indeed built in just such a way, for every
cell of our bodies carries the complete information describing the entire
creature, in the form of DNA.

Three Spheres II |

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 *Three Spheres II* makes use of the
reflective properties of a spherical mirror. Here, as Hofstatder noted, "every
part of the world seems to contain, and be contained in, every other

And so we end where we began, with a self portrait: the work a reflection of the artist, the artist reflected in his work.

# Conclusion

We 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/