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reed magazine logoMarch 2010

Capturing Infinity continued


Despite its simplistic motif, CLII (1959) represented an artistic breakthrough: Escher was now able to construct variations of Coxeter’s figures.

The Miraculous Draught of Fishes

Six months after his breakthrough with CLII,
Escher produced the more sophisticated CLIII,
The Miraculous Draught of Fishes. (1959).

One may ask why Coxeter would send Escher a pattern featuring sevenfold symmetry, even if merely to serve as an analogy. Such a pattern cannot be constructed with straightedge and compass. It could only cause confusion for Escher.

However, Coxeter did present, though very briefly, the principle that Escher sought. I have displayed the essential sentence in italics. In due course, I will show that the sentence holds the key to deciphering Coxeter’s figure. Clearly, Escher did not understand its significance at that time.

On February 15, 1959, Escher wrote again, in frustration, to his son George:

Coxeter’s letter shows that an infinite number of other systems is possible and that, instead of the values 2 and 3, an infinite number of higher values can be used as a basis. He encloses an example, using the values 3 and 7 of all things! However, this odd 7 is no use to me at all; I long for 2 and 4 (or 4 and 8), because I can use these to fill a plane in such a way that all the animal figures whose body axes lie in the same circle also have the same “colour,” whereas, in the other example (CLI), 2 white ones and 2 black ones constantly alternate. My great enthusiasm for this sort of picture and my tenacity in pursuing the study will perhaps lead to a satisfactory solution in the end. Although Coxeter could help me by saying just one word, I prefer to find it myself for the time being, also because I am so often at cross purposes with those theoretical mathematicians, on a variety of points. In addition, it seems to be very difficult for Coxeter to write intelligibly for a layman. Finally, no matter how difficult it is, I feel all the more satisfaction from solving a problem like this in my own bumbling fashion. But the sad and frustrating fact remains that these days I’m starting to speak a language which is understood by very few people. It makes me feel increasingly lonely. After all, I no longer belong anywhere. The mathematicians may be friendly and interested and give me a fatherly pat on the back, but in the end I am only a bungler to them. “Artistic” people mainly become irritated.

Escher’s enthusiasm and tenacity did indeed prove sufficient. Somehow, dur­ing the following months, he taught himself, in terms of the straightedge and the compass, to construct not only Coxeter’s figure but at least one variation of it as well. In March 1959, he completed the second of the woodcuts in his Circle Limit Series.

The simplistic design of the work suggests that it may have served as a practice run for its successors. In any case, Escher spoke of it in humorous terms:

Really, this version ought to be painted on the inside sur­face of a half-sphere. I offered it to Pope Paul, so that he could decorate the inside of the cupola of St. Peter’s with it. Just imagine an infinite number of crosses hanging over your head! But Paul didn’t want it.

In December 1959, he completed the third in the series, the intriguing CLIII, titled The Miraculous Draught of Fishes.

He described the work eloquently, in words that reveal the craftsman’s pride of achievement:

In the colored woodcut Circle Limit III the shortcomings of Circle Limit I are largely eliminated. We now have none but “through traffic” series, and all the fish belonging to one series have the same color and swim after each other head to tail along a circular route from edge to edge. The nearer they get to the center the larger they become. Four colors are needed so that each row can be in complete contrast to its surroundings. As all these strings of fish shoot up like rockets from the infinite distance at right angles from the boundary and fall back again whence they came, not one single component reaches the edge. For beyond that there is “absolute nothingness.” And yet this round world cannot exist without the emptiness around it, not simply because “within” presupposes “without,” but also because it is out there in the “nothingness” that the center points of the arcs that go to build up the framework are fixed with such geometric exactitude.

As I have noted, Escher completed the last of the Circle Limit Series, CLIV, in July 1960. Of this work, he wrote very little of substance:

Here, too, we have the components diminishing in size as they move outwards. The six largest (three white angels and three black devils) are arranged about the center and radiate from it. The disc is divided into six sections in which, turn and turn about, the angels on a black background and then the devils on a white one, gain the upper hand. In this way, heaven and hell change place six times. In the intermediate “earthly” stages, they are equivalent.
reed magazine logoMarch 2010