Escher and Infinity (2/5)
In the previous post, I talked about following Escher’s artistic journey to his moment of great inspiration. Bereft of beautiful surroundings as he moves away from picturesque southern Italy, Escher seeks inspiration inside his mind. He adapts the tessellations that he has seen at the Alhambra palace, made by the Moors in medieval Spain, to create something completely unique. You can read the full post here.
Below are some of his most famous works relating to tiles and tessellated surfaces.
From Tiles to Infinity
Escher takes a very interesting journey. You can almost see the wheels turning in his mind. Many years pass between using tessellations, and his attempting to express infinity. In this time, he produces many sketches and prints. They relate to his thoughts around symmetry, reflections and the application of geometric shapes to show natural objects. He is fascinated by the concept of regular divisions on a plane surface, and spends a lot of time deciphering the different structures that can be formed out of it. Although they do not bring him great fame, his work is noted and appreciated by an unlikely audience - scientists and mathematicians.
‘Metamorphosis’
We must keep in mind that Escher is just a passionately curious woodcut print-maker, with no training in “technical sciences”. The journey he undertakes to discover and accurately portray infinity is no less elegant or inevitable than a mathematical solution.
In his interest to discover different forms of perfectly joined tiles, Escher chances upon crystallography and the molecular structures of crystals. He reads scholarly papers on various molecular symmetry groups, but only goes so far as ‘understanding’ the illustrations. These illustrations, then, become his way to move forward. Since he cannot learn explicitly, he will learn implicitly.
It is interesting to consider how tiles - the smallest individual components - spark his interest in the ‘bigger picture’. From working his way down to basic building blocks, Escher finds ideas to show the entire universe itself.
“When the doors of perception are cleansed, everything appears to man as it truly is - infinite.” - Aldous Huxley
To Escher, infinity begins as a place of infinite smallness and infinite density. In his first explorations, he portrays this “point of infinite smallness” in different ways. Of course, he has no idea what infinity should look like. He is simply following the steps that make sense to him, with the tools at his disposal. Quite accidentally, perhaps, he is following a scientific approach of repeated iterations and incremental progress.
A sphere with two poles and a network of longitudinal and latitudinal circles. Fish swim in a spiral from pole to pole, in alternate rows of black and white. (1958)
Rarely does art subject itself to any objective scale, but through his mathematically ‘elegant’ pursuits, Escher is able to hit upon an objective measure of beauty, symmetry and logic.
He is not looking for the way things looks most pretty, but the way it must inevitably be. He is looking for truth.
‘Path of Life II’ (1958)
But Escher knows that he has not yet found infinity. He is a ruthless self-critic. To him, these points of “infinite smallness” are an illogical limit. Instead of truly representing infinity, they represent the end of the ability of the printmaker or his tools to carry on. He finds the outer edges of his works arbitrary as well, since they do not come to a logical, or inevitable, solution.
His frustrations are to be resolved by communicating with the mathematician Coxeter, who is a fan of Escher’s work. He shares a paper with Escher which introduces the artist to the Poincare Disk, a mathematical model to represent hyperbolic geometry on a 2 dimensional surface. This makes as much sense to Escher as that sentence did to you or I. He complains about not understanding the language of mathematics, even though he continues to find admirers of his art from that line of work. He begs them for simpler explanations, but cannot understand those either. Hilariously, he confesses to his son that although he gets to hear exactly what mathematicians think about his work, he understands none of it.
But he does see something in the Poincare Disk that he has been looking for - a logical way to represent infinity. He is able to learn some valuable things from Coxeter, including how to keep diminishing regular shapes in the direction of the limits.
‘Circle Limit I’ (1958) - the first of a series made by what Escher called “Coxetering”
Escher writes,
“A diminution in the size of the figures … from within outwards, leads to more satisfying results. The limit is no longer a point, but a line which borders the whole complex and gives it a logical boundary.”
Steady work in this direction leads him to create 3 more prints in the Circle Limit series, and the Square Limit print. He shares the grid breakdown of this the Square Limit with Coxeter.
‘Square Limit’ (1964)
‘Circle Limit III’ (1959), a favorite of Coxeter’s
Art for art’s sake, not math
It takes tremendous hardwork and painstaking care to accomplish mathematical accuracy with the basic geometry tools Escher possesses. But his use of symbolism is what makes his work truly remarkable.
Let us look at Circle Limit IV to understand this.
‘Circle Limit IV’ (1960), also known as ‘Heaven and Hell’
The symbols are angels (in white) and devils (in black). They are representative of the forces of good and evil. We see them alternating, fitting together, and reproducing themselves in steadily diminishing size, as they radiate outwards. The six largest (three angels and three devils) pivot about the central point. In this way they create six sections of the circle, where angels and devils alternatively have the upper hand. Repeated use of symbols to convey an idea constitutes motif.
Where the angels are dominant, the black spaces are featureless, and where the devils dominate the white spaces are blank. This creates the theme of heaven and hell with alternating spaces for each. In the intermediate, “earthly” stages, the motifs are equally represented.
Escher’s work inspires me because it shows how far we can reach with dogged curiosity and confidence in one’s vision. He had something, and he worked with it in his own way, with his own tools, to find his own answers. His work earned the validation of the scientific community, and from them he took ideas and played with them as well as he could.
I hope this was an interesting read for you. It was very educational for me to read through the texts and collate the information. If there’s anything you’d like me to touch upon, let me know. If there’s anything I can explain better, let me know!
In the next part of this series, I will talk about MC Escher’s most famous style - his impossible geometries.
