\documentclass{report}
\begin{document}
\begin{flushleft}
Sara Smollett \\
December 17, 1997 \\
Math 23 \\ \bigskip
{\bf Mathematics and the Aesthetic: \\
Hyperbolic Geometry in the Works of M.C. Escher}
\end{flushleft}
\bigskip
The union between mathematics and art is a deep one, but perhaps it
is best illustrated in the works of M.C. Escher. His art studies
included drawing lessons by F.W. van der Haagen and three years of
study at the school of Architecture and Ornamental Design in
Haarlem. He later settled in Rome and made many study-tours through
Italy and Spain where he was influenced by the works he saw,
including the Alhambra. His formal math training was extremely
limited, and he repeatedly denied any understanding of mathematics.
Yet his independent studies and artistic intuitions imply a greater
understanding than he ever admitted to. In attempting to
cogno-intellectualize Escher's artwork, mathematicians have found
that his grasp of mathematics included an understanding of
isometries, symmetry groups, crystallography, chromatic groups, and
tesselation in spherical and hyperbolic geometry.
Several of his works, including his ``Circle Limits'' use the
Poincar\'{e}
disk model of hyperbolic geometry. In this model lines are diameters
and arcs perpendicular to the boundary of a circle at infinity.
Distances appear distorted and angles are preserved. If $1/n+1/k<1/2$,
the Shl\"{a}fli symbol $\{n,k\}$ denotes a regular tessellation of $n$-gons,
where $k$ $n$-gons meet at any vertex. A quasiregular tessellation is
built from two kinds of regular polygons. Every regular tessellation
$\{n,k\}$ can give rise to a quasiregular tessellation quasi-$\{n,k\}$ by
connecting the midpoints of the edges of the regular tessellation.
The problem of regularly dividing the plane interested Escher
greatly. He wrote: ``I cannot imagine what my life would be like if
this problem had never occurred to me. One might say that I am head
over heels in love with it, and I still don't know why.''$^{1}$ In the
Euclidean plane there are seventeen essentially different ``wallpaper''
patterns using combinations of translations, rotations, reflections,
and glide-reflections. The fifteen of these that were discovered and
used by Escher are illustrated in Figure 1 on the next page.
Escher was greatly influenced by the geometer H. S. M. Coxeter.
Escher met him at one of the International Congresses of
Mathematicians in 1954 and soon after asked for an explanation of
how to construct a series of objects that decrease in size as they
reach the boundary of a circle. Coxeter wrote an article for the
Royal Society of Canada on symmetry which included a picure of a
Poincar\'{e} tesselation of $30^{\circ}$, $60^{\circ}$, $90^{\circ}$
triangles. Escher then came
across the idea of a hyperbolic plane in 1958 from a figure in ``A
Symposium on Symmetry'' sent to him by Coxeter. See Figure 2.
Hyperbolic tilings are used in many of Escher's works to create the
effect of a figure getting smaller and smaller while preserving
angles. His works using the Poincar\'{e} model are perhaps the most
pleasing, but he also experimented with rectangular regions and
spirals. When using animals for tiling figures, their backbones form
the basis of the spiral or the lines on non-Euclidean surfaces.
``Smaller and Smaller I'' (Figure 3) is not a spiral, but a tiling
with four lizard heads meeting at each point. The figures are
shrinking geometrically towards the center with concentric rings of
black lizards separated by alternatingly facing lizards.
Hyperbolic tilings appealed to Escher because he liked the idea of
similar (not congruent in Euclidean geometry) figures. In ``Square
Limit'' (Figure 4) the repetition is combined with a reduction to half
size. Although it is easy to imagine many simple ways of creating
such tilings of the plane, no {\it non-trivial} ones have been devised
except the one used by Escher in ``Square Limit.'' The tiles are
bounded by four arcs, the first two forming two sides of a
$45^{\circ}-45^{\circ}-45^{\circ}$ triangle and the other two reduced in ratio
$1:\sqrt{2}$, and
placed so that together they are the ``hypotenuse'' of the triangle.$^{2}$
``Candle'' (Figure 5) was done much earlier than Escher's ``Circle
Limits'' and appears to anticipate his use of the Poincar\'{e} disk
model. The lines are not hyperbolic, but they do bear a striking
resemblence to lines in the Poincar\'{e} model. This work is
mathematical in other ways though. It solves the problem of Tammes,
the packing of circular disks on the surface of a sphere.
Each half of ``Whirlpools'' (Figure 6) and ``Path of Life I and III''
(Figures 7 and 8) are spiral progressions getting smaller and smaller
on the inside. This can be seen in the following manner: start with
a tiling of a plane by congruent tile, then roll up a strip of that
tiling to create a tiling of an infinite circular cylinder. The
projection looking down the axis of the cylinder of this onto a plane
gives the desired spiral.$^{3}$
One of Escher's other works that employs the Poincar\'{e} disk model is
``Butterflies'' (Figure 9). Because the dividing line between the
front and the back wings of a butterfly is perpendicular to its
body, the framework of butterflies can be seen circles intersecting
at right angles as in figure 11.$^{4}$ Similarly, a net of circles only
with six fold symmetry instead of eight fold is used for ``Snakes''
(Figure 10). Each hexagon is surrounded entirely by octagons,
producing the quasi-regular tessellation quasi-$\{6,8\}$ shown in figure
12.$^{5}$
But it was in his ``Circle Limits,'' that Escher felt the greatest
sense of acheivement. ``I do this with the strange feeling that this
piece of work is a `milestone' in my development, but that no one but
myself will ever realize it.''$^{6}$ This ``milestone'' is the use of the
Poincar\'{e} disk model in art.
\begin{quotation}
``{\it Circle Limit I}, being a first attempt, displays all sorts of
shortcomings \ldots and leaves much to be desired\ldots . There is no
continuity, no ``traffic flow'' nor unity of colour in each row\ldots .
In the coloured woodcut {\it Circle Limit III}, the short comings of
{\it 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 colour and swim after each other head to tail along a circular
route from edge to edge\ldots . Four colours are needed so that each
row can be in complete contrast to its surroundings.$^{7}$
\end{quotation}
``Circle Limits I'' and ``IV'' use lines for the backbones of the fish.
In ``Circle Limit III'', the arcs of the backbones cross at angles
of $60^{\circ}$ since there are three at each vertex. Thus, if these were
lines, the triangles would by Euclidean. The arcs actually meet the
circumfrence of the outside circle at angles of about $80^{\circ}$, not
$90^{\circ}$, so they are equidistant curves.$^{8}$
The mathematics involved in creating this tiling are amazingly
complicated. Escher did the entire drawing armed only with simple
drawing instruments and his artist's eyes. Coxeter was astonished by
Escher's precision: ``He got it absolutely right to the millimetre,
absolutely to the millimetre.''$^{9}$ Coxeter derived the same results
using the following trigonometry and figure 19.
Assumptions: the relevant arcs of circles cross each other at
angles of $60^{\circ}$, the regions are quadrangles surrounded by triangles,
and they all meet the boundary of a unit circle at angles $\omega$
and $\phi-\omega$ where $\omega$ is the acute angle on the side of the arc where the regions
are quadrangles.
$$
\omega = \arccos(\sinh(\frac{1}{4}\log2)) \approx 79^{\circ}58'
$$
This result can be derived by more elementary procedure.
Applying the law of cosines to triangle $X_{1}AO_{1}$ yields
$|AO_{1}|^{2} = 1 + |O_{1}X_{1}|^{2} - 2\cos{\omega} |O_{1}X_{1}|$
and similar expressions for triangles $X_{2}AO_{2}$ and $X_{3}AO_{3}$.
Because the angle between two intersecting circles equals the angle
between their radii to a common point, the triangle $O_{1}AC$ has angles
$2\pi/3$, $\pi/4$, and $\pi/12$ as in figure 20. By the law of sines,
$$
\frac{|AO_{1}|}{\sin{2\pi/3}} = \frac{|CO_{1}|}{\sin{\pi/4}} =
\frac{|AO_{2}|-|O_{2}X_{2}|}{\sin{\pi/12}}
$$
Triangle $O_{2}AB$ is similar to triangle $O_{1}AC$ so
$$
\frac{|AO_{2}|}{\sin{2\pi/3}} = \frac{|O_{2}X_{2}|}{\sin{\pi/4}}
$$
So for $v=1$ and $v=2$, $|AO_{v}|^{2} = 3/2|O_{v}X_{v}|^{2}$ which yields the quadratics
$|O_{1}X_{1}|^{2} + 4\cos{\omega}|O_{1}X_{1}| - 2 = 0$ and
$|O_{2}X_{2}|^{2} + 4\cos{\omega}|O_{2}X_{2}| - 2 = 0$.
$|O_{1}X_{1}| = -2\cos{\omega} + \sqrt{4\cos{^{2}\omega}+2} and
|O_{2}X_{2}| = 2\cos{\omega} + \sqrt{4\cos{^{2}\omega}+2}$. Let
$x=2\cos{\omega}$.
From our results using the law of sines, we have
$(\sqrt{3}-1)|O_{1}X_{1}| = 2(|AO_{2}| - |O_{2}X_{2}|) =
\sqrt{6}-2)|O_{2}X_{2}|$.
Combining this with the previous results:
$(\sqrt{3}-1)(-x+\sqrt{x^{2}+2})=(\sqrt{6}-2)(x+\sqrt{x^{2}+2})$.
Thus, $x=2\sinh{\frac{1}{4}\log2}$.
We can then solve for the distances in Circle Limit III.
$|O_{1}X_{1}| \approx 1.10816$, $|AO_{1}| \approx 1.3572$,
$|O_{2}X_{2}| \approx 1.8048$, $|AO_{2}| \approx 2.2104$,
$|O_{2}X_{2}| \approx 0.3376$, and $|AO_{2}| \approx 0.9982$
which agree with Escher's actual measurements.$^{10}$
So was M.C. Escher a mathematician? Escher wrote: ``\ldots I have
often felt closer to people who work scientifically (though I
certainly do not do so myself) than to my fellow artists.''$^{11}$ Many
Escher admires suspect he had more mathematical talent than he was
willing to admit. But Coxeter and others believe he was guided
almost solely by the aesthetic, which is of course closely related to
the mathematic. ``[He was a]bsolutely unaware [of the mathematics
behind `Circle Limit III'. In his own words: `\ldots 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.'{''}$^{12}$
His works correspond with those of crystallographers, often
surpassing their insights. Yet there is a difference in their
motivation to study symmetries. ``[T]hey have opened the gate leading
to an extensive domain, but they have not entered this domain
themselves. By their very nature they are more interested in the
way in which the gate is opened than in the garden lying behind
it.''$^{13}$
In his {\it Regelmatige vakverdeling} (Regular Division of the Plane)
Escher writes:
\begin{quotation}
At first I had no idea at all of the possibility of systematically
building up my figures. I did not know any ``ground rules'' and tried,
almost without knowing what I was doing, to fit together congruent
shapes that I attempted to give the form of animals. Gradually,
designing new motifs became easier as a result of my study of the
literature on the subject, as far as this was possible for someone
untrained in mathematics, and especially as a result of my putting
forward my own layman's theory, which forced me to think through the
possibilities. It remains an extremely absorbing activity, a real
mania to which I have become addicted, and from which I sometimes
find it hard to tear myself away.$^{14}$
\end{quotation}
Escher did not have mathematical training nor did he understand the
ideas behind hyperolic geometry. But he did have an eye for beauty
and a gift of knowing what would look good. He saw that symmetry was
pleasing to the eye, and that regular divisions involved
mathematics. In this sense, he was not a geometer, but his interests
were those of a pure mathematician. He developed mathematical
principles on his own in an effort to understand this beauty. I have
to concur with Gr\"{u}nbaum that ``it is very likely that Escher did not
wish to learn any of the mathematics we think might have helped him,
and that we are much richer for it.''$^{15}$
\bigskip \bigskip
\begin{flushleft}
{\bf Works Cited}
Coxeter, H.S.M. ``Coloured Symmetry.'' {\it M.C. Escher: Art and Science}.
Ed. Coxeter, Emmer, Penrose, Teuber. Amsterdam: Elsevier Science
Publishers B.V., 1988. 15-33.
\\ \medskip
---. ``The Trigonometry of Escher's Woodcut `Circle Limit III'.''
{\it The Mathematical Intelligencer} (1996, v.18, n.4): 42-46.
\\ \medskip
Dunham, Douglas J. ``Creating Hyperbolic Escher Patterns.'' {\it M.C.
Escher: Art and Science}. Ed. Coxeter, Emmer, Penrose, Teuber.
Amsterdam: Elsevier Science Publishers B.V., 1988. 241-248.
\\ \medskip
Ernst, Bruno. {\it The Magic of M.C. Escher}. New York: Barnes \& Noble,
1994.
\\ \medskip
Gr\"{u}nbaum, Branko. ``Mathematical Challenges in Escher's Geometry.''
{\it M.C. Escher: Art and Science}. Ed. Coxeter, Emmer, Penrose, Teuber.
Amsterdam: Elsevier Science Publishers B.V., 1988. 53-67.
\\ \medskip
Hargittai, Istv\'{a}n. ``Lifelong Symmetry: A Conversation with H. S. M.
Coxeter.'' {\it The Mathematical Intelligencer} (1996, v.18, n.4): 38-39.
\\ \medskip
Locher, J.L. et al. {\it M.C. Escher: His Life and Complete Graphic
Work}. Harry N Abrams: New York, 1982.
\\ \medskip
Rigby, J. F. ``Butterflies and Snakes.'' {\it M.C. Escher: Art and
Science}. Ed. Coxeter, Emmer, Penrose, Teuber. Amsterdam: Elsevier
Science Publishers B.V., 1988. 211-220.
\\ \medskip
Strauss, Stephen. ``Art is Math is Art for Professor Coxeter.'' {\it The
Globe and Mail, Canada's National Newspaper}. May 9, 1996.
{\it http://www.math.toronto.edu/\~{}coxeter/art-math.html}.
\end{flushleft}
\bigskip
(Footnotes and Figures available upon request.)
\end{document}