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reed magazine logoMarch 2010

Capturing Infinity continued

workshop drawing

Workshop drawing

Immediately, Escher saw in the figure a realistic method for achieving his goal: to capture infinity. For a suitable motif, such as an angel or a devil, he might create, in method logically precise and in form visually pleasing, infinitely many modified copies of the motif, with the intended effect that the multitude would pack neatly into a disk.

With straightedge and compass, Escher set forth to analyze the figure. The following diagram, based upon a workshop drawing, suggests his first (no doubt empirical) effort:

Escher recognized that the figure is defined by a network of infinitely many circular arcs, together with certain diameters, each of which meets the circular boundary of the ambient disk at right angles. To reproduce the figure, he needed to determine the centers and the radii of the arcs. Of course, he recognized that the centers lie exterior to the disk.

Failing to progress, Escher set the project aside for several months. Then, on November 9, 1958, he wrote a hopeful letter to his son George:

figures b and ab

Escher based the design of CLI on Figure B, which he derived from Figure A. The two are superimposed in Figure AB.

I’m engrossed again in the study of an illustration which I came across in a publication of the Canadian professor H.S.M. Coxeter . . . I am trying to glean from it a method for reducing a plane-filling motif which goes from the center of a circle out to the edge, where the motifs will be infinitely close together. His hocus-pocus text is of no use to me at all, but the picture can probably help me to produce a division of the plane which promises to become an entirely new variation of my series of divisions of the plane. A regular, circular division of the plane, logically bordered on all sides by the infinitesimal, is something truly beautiful.

Soon after, by a remarkable empirical effort, Escher succeeded in adapting Coxeter’s figure to serve as the underlying pattern for the first woodcut in his Circle Limit Series, CLI (November 1958).

One can detect the design for CLI in the following Figure B, closely related to Figure A:


However, Escher had not yet found the principles of construction that un­derlie Figures A and B. While he could reproduce the figures empirically, he could not yet construct them ab initio, nor could he construct variations of them. He sought Coxeter’s help. What followed was a comedy of good inten­tion and miscommunication. The artist hoped for the particular, in practical terms; the mathematician offered the general, in esoteric terms. On December 5, 1958, Escher wrote to Coxeter:

Though the text of your article on “Crystal Symmetry and its Generalizations” is much too learned for a simple, self-made plane pattern-man like me, some of the text illustrations and especially Figure 7, [that is, Figure A] gave me quite a shock.

Since a long time I am interested in patterns with “motifs” getting smaller and smaller till they reach the limit of infinite smallness. The question is relatively simple if the limit is a point in the center of a pattern. Also, a line-limit is not new to me, but I was never able to make a pattern in which each “blot” is getting smaller gradually from a center towards the outside circle-limit, as shows your Figure 7.

I tried to find out how this figure was geometrically con­structed, but I succeeded only in finding the centers and the radii of the largest inner circles (see enclosure). If you could give me a simple explanation how to construct the following circles, whose centers approach gradually from the outside till they reach the limit, I should be immensely pleased and very thankful to you! Are there other systems besides this one to reach a circle-limit?

Nevertheless I used your model for a large woodcut (CLI), of which I executed only a sector of 120 degrees in wood, which I printed three times. I am sending you a copy of it, together with another little one (Regular Division VI ), illustrating a line-limit case.

On December 29, 1958, Coxeter replied:

I am glad you like my Figure 7, and interested that you succeeded in reconstructing so much of the surrounding “skeleton” which serves to locate the centers of the circles. This can be continued in the same manner. For instance, the point that I have marked on your drawing (with a red • on the back of the page) lies on three of your circles with centers 1, 4, 5. These centers therefore lie on a straight line (which I have drawn faintly in red) and the fourth circle through the red point must have its center on this same red line.

In answer to your question “Are there other systems be­sides this one to reach a circle limit?” I say yes, infinitely many! This particular pattern [that is, Figure A] is denoted by {4, 6} because there are 4 white and 4 shaded triangles coming together at some points, 6 and 6 at others. But such patterns {p, q} exist for all greater values of p and q and also for p = 3 and q = 7,8,9,... A different but related pattern, called <<p, q>> is ob­tained by drawing new circles through the “right angle” points, where just 2 white and 2 shaded triangles come together. I enclose a spare copy of <<3, 7>>… If you like this pattern with its alternate triangles and heptagons, you can easily derive from {4, 6} the analogue <<4, 6>>, which consists of squares and hexagons.
Regular Division

Regular Division VI (1957) illustrates Escher’s ability to execute a line limit.


Escher completed CLI, the first in the
Circle Limit Series, in 1958.

reed magazine logoMarch 2010