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march 2010
Capturing Infinity
Reed mathematics professor Thomas Wieting explores
the hyperbolic geometry of M.C. Escher’s
Angels and Devils.
Capturing Infinity
The Circle Limit Series of M.C. Escher
BY THOMAS WIETING
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.
The Dutch artist
Maurits C. Escher
(1898–1972)
Escher completed
CLIV,
also
known as
Angels and Devils,
in 1960.
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March 2010
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.
Day and Night
(1938) is the most popular of Escher’s works.
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.
would not permit what the real materials of
his workshop 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.
In 1954, the organizing committee for the
International Congress of Mathematicians
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.” Three years
later, while writing an article on symme-
try to serve as the presidential address to
the Royal Society of Canada, the eminent
mathematician H.S.M. Coxeter recalled the
exhibition. He wrote to Escher, requesting
permission 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 publish reproductions of my regular-
flat-fillings, but I am also proud of your inter-
est in them!
Serendipity
Figure A
The 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-rising level of technical skill. However,
following the transformation, Escher produced
an inspired stream of the utterly original works
that are now identified 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.
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:
However, while immensely pleased in prin-
ciple, Escher was dissatisfied in practice with
a particular feature of the tessellations. He
found that the logic of the underlying patterns
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:
Workshop drawing
Escher recognized that the figure is defined
by a network of infinitely many circular arcs,
March 2010
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Capturing Infinity
continued
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 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.
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.
Frustration
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 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 intention and miscommunication.
The 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 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 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
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 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 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 con-
sists of squares and hexagons.
•
Figure B
Figure AB
Escher based the design of
CLI
on Figure B,
which he derived from Figure A.
The two are superimposed in Figure AB.
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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is “absolute nothingness.” And yet this round
world cannot exist without the emptiness
around it, not simply because “within” presup-
poses “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.
Despite its simplistic motif,
CLII
(1959) represented an
artistic breakthrough: Escher was now able to construct
variations of Coxeter’s figures.
Six months after his breakthrough with
CLII,
Escher produced the more sophisticated
CLIII,
The Miraculous Draught of Fishes.
(1959).
ver y briefly, the principle that Escher
sought. I have displayed the essential sen-
tence in italics. In due course, I will show
that the sentence holds the key to decipher-
ing 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 num-
ber of other systems is possible and that,
instead of the values 2 and 3, an infinite num-
ber 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 per-
haps 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 pur-
poses 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 mathemati-
cians 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.
Success
Escher’s enthusiasm and tenacity did indeed
prove sufficient. Somehow, during the fol-
lowing months, he taught himself, in terms
of the straightedge and the compass, to con-
struct 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 sug-
gests 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 hang-
ing over your head! But Paul didn’t want it.
Perhaps Escher intended that this woodcut
should inspire not commentary but con-
templation.
Remarkably, while
CLI
and
CLIV
are
based upon Figures A and B,
CLII
and
CLIII
are based upon the following subtle varia-
tions of them:
Figure C
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 elimi-
nated. 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 com-
ponent reaches the edge. For beyond that there
Figure D
Figure CD
March 2010
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