Note: this paper is much better with the figures and more reasonable mathematics notation.
Mathematics and the Aesthetic:
Hyperbolic Geometry in the Works of M.C. Escher
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 studytours 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
cognointellectualize 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 Poincaré 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 Shläfli symbol {n,k} denotes a regular tessellation of ngons, where k ngons 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 glidereflections. 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 Poincaré tesselation of 30^{ƒ}, 60^{ƒ}, 90^{ƒ} 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 Poincaré 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 nonEuclidean 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 nontrivial 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^{ƒ}45^{ƒ}45^{ƒ} triangle and the other two reduced in ratio 1:÷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 Poincaré disk model. The lines are not hyperbolic, but they do bear a striking resemblence to lines in the Poincaré 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 Poincaré 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 quasiregular 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 Poincaré disk model in art.
``Circle Limit I, being a first attempt, displays all sorts of shortcomings ... and leaves much to be desired... . There is no continuity, no ``traffic flow'' nor unity of colour in each row... . In the coloured woodcut Circle Limit III, the short comings 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 colour and swim after each other head to tail along a circular route from edge to edge... . Four colours are needed so that each row can be in complete contrast to its surroundings.^{7}
``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^{ƒ} 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^{ƒ}, not 90^{ƒ}, 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^{ƒ}, the regions are quadrangles surrounded by triangles, and they all meet the boundary of a unit circle at angles w and fw where w is the acute angle on the side of the arc where the regions are quadrangles.

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}  2cosw 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 2p/3, p/4, and p/12 as in figure 20. By the law of sines,

Triangle O_{2}AB is similar to triangle O_{1}AC so

So for v = 1 and v = 2, AO_{v}^{2} = 3/2O_{v}X_{v}^{2} which yields the quadratics O_{1}X_{1}^{2} + 4coswO_{1}X_{1}  2 = 0 and O_{2}X_{2}^{2} + 4coswO_{2}X_{2}  2 = 0. O_{1}X_{1} = 2cosw + ÷{4cos^{2}w+2} and O_{2}X_{2} = 2cosw + ÷{4cos^{2}w+2}. Let x = 2cosw.
From our results using the law of sines, we have (÷31)O_{1}X_{1} = 2(AO_{2}  O_{2}X_{2}) = ÷62)O_{2}X_{2}.
Combining this with the previous results: (÷31)(x+÷{x^{2}+2}) = (÷62)(x+÷{x^{2}+2}). Thus, x = 2sinh^{1}/_{4}log2.
We can then solve for the distances in Circle Limit III. O_{1}X_{1} ª 1.10816, AO_{1} ª 1.3572, O_{2}X_{2} ª 1.8048, AO_{2} ª 2.2104, O_{2}X_{2} ª 0.3376, and AO_{2} ª 0.9982 which agree with Escher's actual measurements.^{10}
So was M.C. Escher a mathematician? Escher wrote: ``... 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: `... 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 Regelmatige vakverdeling (Regular Division of the Plane) Escher writes:
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}
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 Grünbaum 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}
Works Cited
Coxeter, H.S.M. ``Coloured Symmetry.'' M.C. Escher: Art and
Science.
Ed. Coxeter, Emmer, Penrose, Teuber. Amsterdam: Elsevier Science
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. ``The Trigonometry of Escher's Woodcut `Circle Limit III'.''
The Mathematical Intelligencer (1996, v.18, n.4): 4246.
Dunham, Douglas J. ``Creating Hyperbolic Escher Patterns.'' M.C.
Escher: Art and Science. Ed. Coxeter, Emmer, Penrose, Teuber.
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Ernst, Bruno. The Magic of M.C. Escher. New York: Barnes &
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Grünbaum, Branko. ``Mathematical Challenges in Escher's Geometry.''
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Hargittai, István. ``Lifelong Symmetry: A Conversation with H. S.
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Locher, J.L. et al. M.C. Escher: His Life and Complete Graphic
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Rigby, J. F. ``Butterflies and Snakes.'' M.C. Escher: Art and
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Strauss, Stephen. ``Art is Math is Art for Professor Coxeter.'' The Globe and Mail, Canada's National Newspaper. May 9, 1996. http://www.math.toronto.edu/~coxeter/artmath.html.
(Footnotes and Figures available upon request.)