In July 1960 Escher completed the last of his four ‘circle limits’. He had struggled wit it for a while, but it was a publication by the Canadian professor H.S.M. Coxeter which brought him on the right path. He had met this professor at the University of Toronto in 1954, during the International Congress of Mathematicians. In the article, Coxeter described how a tessellation from the center to the edge of a circle is increasingly reduced and the motifs come to lie infinitely close together. In 1957 Coxeter gave a lecture for the Royal Society of Canada and he asked Escher by letter if he could use a few works of the graphic artist in the lecture. Afterwards, Coxeter sent Escher a print of it (which had been published under the name Crystal Symmetry and Its Generalizations ), in which he also included the figure that Escher would become so enthusiastic about. Coxeter in turn based this figure on the work of the French mathematician Jules Henri Poincaré, who visualized this form of hyperbolic geometry in his Poincaré disc.
Escher got his inspiration mainly from the illustrations, with the accompanying text he couldn’t do much.
‘Coxeter’s hokus-pokus text does nothing for me, but the picture probably helps me to create a tessellation, which promises to be a completely new variant of my series of tilings of the plane. A circular, regular tiling, which is logically bounded on all sides by the infinitely small, is something wondrous. Almost as beautiful as the regular tiling of the sphere. At the same time I have the feeling I’m getting more and more removed from what I can achieve “success” with the “public” with. But what can I do about it, if a problem like this fascinates me so much that I can not get rid of it. It is less easy than it seems. Try it: put one (or four) squares of any size in the middle of a circle (eg separated by two perpendicular diameters) and gradually reduce them to the outside, and so on, that they line up like chess-board fields. You don’t get there with only four-fold axles; you have to alternate the 4-tier with 6-tier in a most curious manner, which is normally impossible on the flat surface. The boundaries are therefore only rectilinear for a small part (only 3 intersecting middle lines) and for the rest they are all circles. Without the image of Coxeter I would never have come up with the idea.’
A correspondence between the graphic artist and the mathematician followed, with Escher asking for advice on how to proceed with his attempts to get a grip on these hyperbolic tilings. Coxeter in his turn learned from Escher because he came up with solutions for this form that the professor hadn’t thought about yet. The mathematical side, however, remained extremely difficult for Escher. In 1960, when he sent the professor a print of Circle Limit III, he received an enthusiastic letter back. But again he did not understand what Coxeter wanted to put in writing.
‘Three pages full of explainings of what I have actually done. Too bad I don’t understand nothing, absolutely nothing, about it.’
As a basis for Circle Limit IV , Escher used a drawing from one of his sketchbooks: Regular division drawing nr. 45, made during Christmas 1941. The subtitle of Circle Limit IV is Heaven and Hell. The print shows white ‘angels’ and black ‘devils’ (in the form of bats). The three angels and three devils in the middle complement each other perfectly. Good and evil need each other to be visible, to infinity.
What’s so fascinating and brilliant about Escher is that he succeeded in making hyperbolic tilings based on complex mathematical theories that many mathematicians have racked their brains about, while he himself does not understand anything about these theories.
Professor Thomas Wieting wrote a very interesting paper about the development of the circle limits. It’s also available as pdf.
In this video Coxeter brings Escher’s circle limits together with the theory behind them.