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.