The story of Escher’s life and work is of a loner, who’s freed himself from the dogma of division between art and science. Impossible as his dreams might look, the very principle that keeps the magical world intact is a concept that has been familiar to mathematicians and crystallographers for centuries – Regular Division of the Plane. Only, as the artist has expressed, they were only interested in ‘opening the gate of the great domain’ by researching the theory and methods for dividing a plane regularly but none has ventured into exploring the vast possibilities that lie beyond that gate. However, Escher’s skill and talent as a graphic artist gave, otherwise abstract idea, a more tangible and concrete reality.

The world Escher portrays in his work is mind-boggling, sci-fi, and, simply, beyond our reality: a waterfall that runs upwards, a tower where you climb on top of it only to realize yourself at the bottom again, weird creatures and clash of dimensions. It can’t get more creative, more irrational than this, yet every detail is a meticulously executed fruit of considerable calculation. Although his works are portals to an alternate dimension, his rationality, clear logic and neat approach still keep our feet on the ground preventing us from drifting off to vagueness.

What is the Regular Division of the Plane?

A plane, which one must imagine as extending without boundaries in all directions, can be filled or divided into infinity, according to a limited number of systems, with similar geometric figures that are contiguous on all sides without leaving “empty spaces”

The simplest way to understand how RDP works is a floor covered in tiles. Each tile piece is of the same form and size. Usually, tiles are in square shape and fit together seamlessly. RDP is the two identical sets of straight parallel lines that intersect each other at an angle of ninety degrees. Using this example as a starting point, one can imagine series going from simplicity to complexity in types of system for division of planes. This can be done by using “tiles” in all sorts of shapes and forms–rectangles, rhombi’s, triangles and hexagons – as long as they satisfy the following conditions:

  • They must have a ‘closed form’ – it must be possible to capture their entire shape within a closed outline; they must be detached objects. This already greatly limits the choice among the infinite number of different forms that surround us.

For example, plants, which figuratively and literally are more strongly attached to the earth than animals, do not lend themselves well to being objects for the division of planes because it is difficult to represent them without showing whatever they attach themselves to. Thus, parts of plants, a flower, or a leaf (unless fallen to the ground) are also not typical figures for the division of planes because they are bound to concepts such as stem, stalk, branch and trunk.

  • The outline that surrounds their form of appearance must be as characteristic as possible. It ought to identify the nature of the object clearly. The effect of the silhouette must preferably be so strong that one recognizes the object even without many internal detail lines, which always act in a disturbing way with regard to the figure as a unity.
  • The outline must have no indentations and bulges that are too shallow or too deep. These also make the figure as a whole difficult to distinguish. The contiguous black and white elements must be easy to separate from each other without too much effort for the viewer’s eye.

Composing such a jigsaw puzzle is done more or less unconsciously. While drawing, I feel as if I were a spiritualistic medium, controlled by the creatures that I am conjuring up, and it is as if they themselves decide on the shape in which they like to appear. – M.C. Escher

Best objects to be represented as a motif for Regular Division of the Plane

Motifs for division of planes that satisfy all three of these conditions are scarcer than one would think. Inanimate objects that are recognizable as a specific and known thing almost always belong in the category of utilitarian manufactured items. Their silhouette is generally too simple, too straight, or bent with too simple a curve; in other words, the line is not capricious enough.

Most useful of all are the shapes of living beings, and are best when the silhouette is shown in the most characteristic a way as possible.  Experience has taught me that, of all living creatures, the silhouette of gliding birds and fishes are the most gratifying forms for playing the game of plane filling. The silhouette of a flying bird has just the required angularity, and the indentations and bulges of the outline are neither excessively nor insufficiently pronounced. In addition, the bird’s aspect is typical both from above and below, as well as from front and side. A fish is hardly less suitable; only from the front its silhouette immediately unserviceable.

Escher on how he started his exploration

I can’t say how my interest in the regular division of the planes originated and whether outside influences had a primary effect on me. My first intuitive step in that direction had already been taken as a student at the School of Architecture and Decorative Arts in Haarlem. This was before I got to know the Moorish majolica mosaics in the Alhambra, which made a profound impression on me. The Moors use exclusively mathematical abstract motifs for their wall and floor coverings in majolica mosaics. I assume that this is because the use of animal and human images as motifs for the decoration of planes is forbidden to them on religious grounds.

After that first Spanish trip in 1922, I became more and more intrigued by the fitting together of congruent figures …What fascinated me the most of all is the double function of the line of separation between two contiguous figures. It is just as indispensable for one motif as it is for the other. Over and over again it was, and still is, a great joy to have to “found” such a motif that repeats itself rhythmically in accordance with a specific system and thus obeys immovable laws. It gives one the sensation of approaching something that is primeval and eternal.

… I initially did not go beyond searching for and sensing laws to which I submitted by necessity without knowing or understanding them. Finally, it was pointed out to me by the scientific community that the regular division of the planes in mathematical, congruent figures is part of the study of geometric crystallography… I began to get an overall picture of the possibilities offered by the regular division of planes. For the first time I dared to do compositions based on this phenomenon. I dared, that is, to work on the problem of expressing unboundedness in an enclosed plane that is bound by specific dimensions, while retaining the characteristic and fascinating rhythm.

Mosaic I, 1951

Is his work Art or Mathematics?

If it is Art, then no art critic or historian talked about this. But the regular division of the plane has more than just a decorative value. Crystallographers have given us a definition of the concept and have researched and determined what and how many systems and methods exist for dividing a plane regularly. By doing this they have opened the gate that gave access to a vast domain, but they themselves have not entered. Their nature is such that they are more interested in the way the gate is opened than in the garden that lies behind it.

I’m walking around there all by myself, in that splendid garden that is in no way my property, the gate of which stands wide open for everyone. I spend time there in refreshing, but also oppressive loneliness… Because what fascinates me, and what I experience as beauty, is apparently considered dull and dry by others.

Collection of lectures, letters and personal memoirs by the ingenious graphic artist M.C Escher. Eloquently written, it offers a great insight into the creative process and his most famous artworks.