
Escher wrote back to Coxeter that this figure “gave me quite a shock,” since it showed him how to design a pattern in which the motifs become ever smaller toward a limiting circle [Co2]. Escher was able to reconstruct the circular arcs in Coxeter's figure and then use them to create his first circle limit pattern, Circle Limit I which he included with his letter to Coxeter. Figure 2 below shows a rough computer rendition of that pattern, with interior detail for a few of the fish. It is easy to see the connection between Figures 1 and 2.

Escher also mentioned in his letter that he had long known of patterns with one internal limit point, and was familiar with patterns with a limiting line. Escher's prints Development II (1939) and Smaller and Smaller (1956) (Catalog numbers 310 and 413 in [Bo1]) and his notebook drawing number 65 (1944) [Sc1] have single limit points; the last two are invariant under a similarity. His prints Regular Division of the Plane VI (1957) and Square Limit (1964) have line limits (Catalog numbers 421 and 443 [Bo1]).
The points of the Poincaré disk model of hyperbolic geometry are the interior points of a bounding circle in the Euclidean plane. In this model, hyperbolic lines are represented by circular arcs that are perpendicular to the bounding circle, including diameters. Figures 1 and 2 show examples of these perpendicular circular arcs. Equal hyperbolic distances are represented by ever smaller Euclidean distances as one approaches the bounding circle. For example, all the triangles in Figure 1 are the same hyperbolic size, as are all the black fish (or white fish) of Figure 2. The patterns of Figures 1 and 2 are closely related to the regular hyperbolic tessellation {6,4} shown in Figure 3 below. In general, {p,q} denotes the regular tessellation by regular psided polygons with q of them meeting at each vertex. David Joyce has a web site Hyperbolic Tessellations that includes regular, quasiregular, and star hyperbolic tessellations [Jo1]. His site has a Java applet that allows users to create their own hyperbolic tessellations. Martin Deraux also has a web site Hyperbolic tessellations with a Java applet that allows users to create different triangle tessellations such as that of Figure 1 [De1]. Don Hatch has a large array of regular hyperbolic tessellations at his web site Hyperbolic Planar Tesselations [Ha1].

In addition to Escher, other artists have created hyperbolic patterns during the past few decades. Some, including Escher, used “classical” straightedge and compass constructions (described in [Go1]). One such artist is Ruth Ross, who created patterns using different kinds of sea shells.
Other artists used computer methods, such as those used to generate the patterns of this essay [Du1]. Helaman Ferguson realized the {7,3} tessellation in the stone base for his sculpture Eight fold Way. He has also created a leather quilt and a printed pattern Big Red 5 using the {5,4} tessellation. For more on Ferguson's work, see his web site http://www.helasculpt.com/gallery/index.html [Fe1]. Irene Rousseau has also used the {5,4} tessellation to create a precise mosaic pattern. Jan Abas used the {6,4} tessellation in his Islamic star pattern, Hyperbolic Mural [Ab1]. Craig Kaplan has written a general program that generates Islamic star patterns in each of the classical geometries [Ka1]. Tony Bomford used a mixture of classical and computer methods in creating several hooked rugs based on the {5,4} and {6,4} tessellations.

Escher quite successfully overcame all of his criticisms of Circle Limit I in his print Circle Limit III, which has 4color symmetry. It is based on the {8,3} tessellation, as is shown in Figure 5 below. Note that the noses and left fin tips of the fish are at alternate vertices of the octagons.

In describing the print to Coxeter, Escher wrote: “As all these strings of fish shoot up like rockets from infinitely far away, perpendicularly from the boundary, and fall back again whence they came, not one single component ever reaches the edge.” ([Co2] page 20). The white backbones of each stream of fish make prominent arcs on the print and it is tempting to guess that these arcs are hyperbolic lines (i.e. circular arcs perpendicular to the bounding circle). Escher's remark might be interpreted to mean this. But this is not the case. As Coxeter discovered, careful measurements of Circle Limit III show that all the white arcs make angles of approximately 80 degrees with the bounding circle. This is correct, since the backbone arcs are not hyperbolic lines, but equidistant curves, each point of which is an equal hyperbolic distance from a hyperbolic line [Co2], [Co3].
In the Poincaré model, equidistant curves are represented by circular arcs that intersect the bounding circle in acute (or obtuse) angles. Points on such arcs are an equal hyperbolic distance from the hyperbolic line with the same endpoints on the bounding circle. For any acute angle and hyperbolic line, there are two equidistant curves (“branches”), one on each side of the line, making that angle with the bounding circle [Gr1]. Equidistant curves are the hyperbolic analog of small circles in spherical geometry. For example, every point on a small circle of latitude is an equal distance from the equatorial great circle; and there is another small circle in the opposite hemisphere the same distance from the equator.
Each of the backbone arcs in Circle Limit III makes the same angle A with the bounding circle. Coxeter [Co2] used hyperbolic trigonometry to show that A is given by the following expression:
One can imagine a threeparameter family (k,l,m) of
Circle Limit III fish patterns
in which k right fins, l left fins, and m noses
meet, where m must be odd so that the fish swim head to tail.
The pattern would be hyperbolic, Euclidean, or spherical depending on
whether
1/k + 1/l + 1/m
is less than, equal to, or greater than 1.
Circle Limit III would be denoted (4,3,3) in this system.
Escher created another pattern in this family, his Euclidean
notebook drawing number 123, denoted (3,3,3), in which each fish
swims in one of three directions [Sc1]. All the fish swimming in one
direction are the same color.
The pattern on the 2003 Math Awareness Month poster is (5,3,3) in this system,
and is shown below in Figure 6.

It is necessary to use six colors to color the lines of fish symmetrically and adhere to the mapcoloring principle: no adjacent fish should be the same color. Following Coxeter's calculation [Co2], it is easy to show that the angle A between the backbones and the bounding circle is given by:
The 2003 Math Awareness Month poster design is just one example of the connection between mathematics and art. Of course there are numerous other connections, including those inspired by Escher in the recent book M.C.Escher's Legacy [Sc2]. My article [Du3] and electronic file on the CD Rom that accompanies that book contain many examples of computergenerated hyperbolic tessellations inspired by Escher's art. For more on Escher's work, see the Official M. C. Escher Web site http://www.mcescher.com/ [Es1].
[Ab1] Abas, S. Jan, Web site: http://www.bangor.ac.uk/~mas009/part.htm
[Bo1] Bool, F.H., Kist, J.R., Locher, J.L., and Wierda, F., editors, M. C. Escher, His life and Complete Graphic Work, Harry N. Abrahms, Inc., New York, 1982. ISBN 0810908581
[Co1] Coxeter, H. S. M., “Crystal symmetry and its generalizations,” Royal Society of Canada(3), 51 (1957), 113.
[Co2] Coxeter, H. S. M., “The nonEuclidean symmetry of Escher's Picture `Circle Limit III',” Leonardo, 12 (1979), 1925, 32.
[Co3] Coxeter, H. S. M., “The Trigonometry of Escher's Woodcut 'Circle Limit III',” The Mathematical Intelligencer, 18 no. 4 (1996) 4246. Updated and corrected version appears in [Sc2] below.
[Co4] Coxeter, H. S. M.,
“Angels and devils,” in
The Mathematical Gardner, David A. Klarner, editor,
Wadsworth International, 1981 (out of print). ISBN 0534980155
Republished as:
Mathematical Recreations: A Collection in Honor of Martin Gardner,
David A. Klarner, editor, Dover Publishers, 1998. ISBN 0486400891
[De1] Deraux, Martin, Interactive tessellation web site: http://www.math.utah.edu/~deraux/tessel/
[Du1] Dunham, D., “Hyperbolic symmetry,” Computers and Mathematics with Applications, Part B 12 (1986), no. 12, 139153.
[Du2] Dunham, D., “Transformation of Hyperbolic Escher Patterns,” Visual Mathematics (an electronic journal), 1, No. 1, March, 1999.
[Du3] Dunham, D., “Families of Escher Patterns,” in [Sc2] below, pp. 286296.
[Es1] Official M. C. Escher Web site, published by the M.C. Escher Foundation and Cordon Art B.V. http://www.mcescher.com/
[Fe1] Ferguson, Helaman, Web site: http://www.helasculpt.com/gallery/index.html
[Go1] GoodmanStrauss, Chaim, “Compass and straightedge in the Poincaré disk,” Amer. Math. Monthly, 108 (2001), no. 1, 3849.
[Gr1] Greenberg, Marvin, Euclidean and NonEuclidean Geometries, 3rd Edition, W. H. Freeman and Co., 1993. ISBN 0716724464
[Ha1] Hatch, Don, Hyperbolic tessellations web site: Hyperbolic Planar Tesselations [Ha1].
[He1] Henderson, David W., and Daina Taimina, Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces, 2nd Ed., Prentice Hall, 2000. ISBN 0130309532 Web link: http://www.mathsci.appstate.edu/~sjg/class/3610/hen.html
[Jo1] Joyce, David, Hyperbolic tessellations web site: http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
[Ka1] Kaplan, Craig S., “Computer generated Islamic star patterns,” Bridges 2000, Mathematical Connections in Art, Music and Science. Winfield, Kansas, USA, 2830 July 2000. ISBN 0966520122 Web link: Abstract and PDF
[Ma1] Magnus, Wilhelm, Noneuclidean Tesselations and Their Groups, Academic Press, 1974. ISBN 0124654509
[Sc1] Schattschneider, Doris, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher, W. H. Freeman, New York, 1990. ISBN 0716721260
[Sc2] Schattschneider, Doris, and Michele Emmer, editors, M. C. Escher's Legacy: A Centennial Celebration, Springer Verlag, 2003. ISBN 354042458X