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Capturing Inﬁnity
‰
march 2010
In July 1960, shortly after his 62nd birthday, the graphic artist M.C. Escher completed
Angels and Devils,
the fourth (and ﬁnal) woodcut in his Circle Limit Series. I have a vivid memory of my ﬁrst view of a print of this astonishing work. Following sensations of surprise and delight, two questions rose in my mind. What is the underlying design? What is the purpose? Of course, within the world of Art, narrowly interpreted, one might regard such questions as irrelevant, even impertinent. However, for this particular work of Escher, it seemed to me that such questions were precisely what the artist intended to excite in my mind. In this essay, I will present answers to the foregoing questions, based upon Escher’s articles and letters and upon his workshop drawings. For the mathematical aspects of the essay, I will require nothing more but certainly nothing less than thoughtful applications of straightedge and compass.
Capturing Inﬁnity
Te Circle Limit Series of M.C. Escher
BY THOMAS WIETING
Escher completed
CLIV
, also known as
Angels and Devils
, in 1960. The Dutch artist Maurits C. Escher (1898–1972)
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March 2010
Capturing Inﬁnity
In 1959, Escher described, in retrospect, a transformation of attitude that had occurred at the midpoint of his career:
I discovered that technical mastery was no longer my sole aim, for I was seized by another desire, the existence of which I had never sus-pected. Ideas took hold of me quite unrelated to graphic art, notions which so fascinated me that I felt driven to communicate them to others.
Te woodcut called
Day and Night,
com-pleted in February 1938, may serve as a sym-bol of the transformation. By any measure, it is the most popular of Escher’s works. Prior to the transformation, Escher pro-duced for the most part portraits, landscapes, and architectural images, together with com-mercial designs for items such as postage stamps and wrapping paper, executed at an ever-ris
ing level of technical skill. However, following the transformation, Escher produced an inspired stream of the utterly srcinal works that are now iden
tiﬁed with his name: the illu-sions, the impossible ﬁgures, and, especially, the regular divisions (called tessellations) of the Euclidean plane into
potentially
inﬁnite populations of ﬁsh, reptiles, or birds, of stately horsemen or dancing clowns.Of the tessellations, he wrote:
This is the richest source of inspiration that I have ever struck; nor has it yet dried up.
However, while immensely pleased in prin-ciple, Escher was dissatisﬁed in prac
tice with a particular feature of the tessellations. He found that the logic of the underlying patterns
Day and Night
(1938) is the most popular of Escher’s works. Figure A
Regular Division III
(1957) demonstrates Escher’s mastery of tessellation. At the same time, he was dissatisﬁed with the way the pattern was arbitrarily interrupted at the edges.
would not permit what the real materials of his work
shop required: a ﬁnite boundary. He sought a new logic, explicitly visual, by which he could organize
actually
inﬁnite populations of his corporeal motifs into a patch of ﬁnite area. Within the framework of graphic art, he sought, he said, to capture inﬁnity.
Serendipity
In 1954, the organizing committee for the International Congress of Mathe
maticians promoted an unusual special event: an exhi-bition of the work of Escher at the Stedelijk Museum in Amsterdam. In the companion catalogue for the exhibition, the committee called attention not only to the mathemati-cal substance of Escher’s tessellations but also to their “peculiar charm.” Tree years later, while writing an article on symme-try to serve as the pres
idential address to the Royal Society of Canada, the eminent mathematician H.S.M. Coxeter recalled the exhibition. He wrote to Escher, requesting per
mission to use two of his prints as illus-trations for the article. On June 21, 1957, Escher responded enthusiastically:
Not only am I willing to give you full permis-sion to pub
lish reproductions of my regular-ﬂat-ﬁllings, but I am also proud of your inter-est in them!
In the spring of 1958, Coxeter sent to Escher a copy of the article he had written. In addi-tion to the prints of Escher’s “ﬂat-ﬁllings,” the article contained the following ﬁgure, which we shall call Figure A: 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. Te following diagram, based upon a workshop drawing, suggests his ﬁrst (no doubt empirical) eﬀort:
Workshop drawing
Escher recognized that the ﬁgure is deﬁned by a network of inﬁnitely many circular arcs,
March 2010
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23
together with certain diameters, each of which meets the circular boundary of the ambient disk at right angles. o 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. Ten, on Novem-ber 9, 1958, he wrote a hopeful letter to his son George:
I’m engrossed again in the study of an illus-tration 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 reduc-ing 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 pro-duce 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 bor-dered 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:
Frustration
However, Escher had not yet found the principles of construction that un
derlie Figures A and B. While he could reproduce the ﬁgures empirically, he could not yet construct them
ab initio,
nor could he con-struct variations of them. He sought Cox-eter’s help. What followed was a comedy of good inten
tion and miscommunication. Te artist hoped for the particular, in prac-tical 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 illustra-tions and especially Figure 7, [that is, Figure A] gave me quite a shock. Since a long time I am interested in pat-terns with “motifs” getting smaller and small-er 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 geo-metrically con
structed, 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 fol-lowing 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
Escher completed
CLI,
the ﬁrst in the Circle Limit Series, in 1958.
Regular Division VI
(1957) illustrates Escher’s ability to execute a line limit.
Capturing Inﬁnity
continued
Escher based the design of
CLI
on Figure B, which he derived from Figure A. The two are superimposed in Figure AB.Figure BFigure AB
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 interest-ed 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
ã
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 be
sides this one to reach a circle limit?” I say yes, inﬁnitely many! This partic-ular 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 pat-tern, called <<p, q>> is ob tained 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 con-sists of squares and hexagons.
One may ask why Coxeter would send Escher a pattern featuring sevenfold sym-metry, 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
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March 2010

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