Math Awareness Month - April 2003

Mathematics and Art


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Artists have created patterns on plane surfaces for millennia, and on spheres for hundreds of years. But it was only recently that the hyperbolic plane has been utilized for artistic purposes, though mathematicians have been drawing hyperbolic patterns for more than 100 years (see [Ma1] for examples). M. C. Escher was most likely the first artist to make use of all three of the classical geometries: Euclidean, spherical, and hyperbolic geometry. In fact he realized his angels and devils pattern in each of these geometries [Co4]. Below, I will trace some of the history of hyperbolic art starting with Escher's hyperbolic inspiration from a figure by the mathematician H. S. M. Coxeter. Then I will discuss some of the hyperbolic art created by other artists, and I will finish by explaining the relationship between Escher's print Circle Limit III and the 2003 Math Awareness Month poster design.

Escher and Hyperbolic Geometry

Coxeter and Escher corresponded after first meeting at the 1954 International Congress of Mathematicians. In 1958 Coxeter sent Escher a letter containing a reprint of Coxeter's paper Crystal Symmetry and Its Generalizations [Co1]. Figure 7 of that paper contained a triangle tessellation of the Poincaré disk model of the hyperbolic plane. That tessellation is reproduced as Figure 1 below.

Figure 1: A tessellation of the hyperbolic plane by 30-45-90 triangles.

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.

Figure 2: A computer rendition of the pattern in Escher's print Circle Limit I, showing interior detail for a few of the fish.

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]).

Hyperbolic Geometry and Hyperbolic Art

Hyperbolic geometry was independently discovered about 170 years ago by János Bolyai, C. F. Gauss, and N. I. Lobatchevsky [Gr1], [He1]. The hyperbolic plane is the only complete surface with constant negative curvature. About 100 years ago David Hilbert essentially proved that there was no smooth isometric embedding of the hyperbolic plane into Euclidean 3-space, so we must rely on mathematical models to view it [He1]. These models must, perforce, distort distances and possibly angles. One such model, the Poincaré disk model, is useful to artists since it is conformal (angles have their Euclidean measure) and it is displayed in a bounded region of the Euclidean plane, so that it can be viewed in its entirety.

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 p-sided polygons with q of them meeting at each vertex. David Joyce has a web site Hyperbolic Tessellations that includes regular, quasi-regular, 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].

Figure 3: The regular tessellation {6,4} of the hyperbolic plane.

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.

Circle Limit III and the Poster Design

Escher had several criticisms of his Circle Limit I pattern. First, the fish are "rectilinear", without the curved outlines of real fish. Also, there is no "traffic flow" along the backbone lines -- the fish change directions after two fish, and the fish change colors along lines of fish ([Co2] page 20). Another criticism, which Escher didn't make, is that the black and white fish are not equivalent (the nose angles are different), so there is no color symmetry. Escher's latter criticisms could be overcome by basing the fish pattern on the {6,6} tessellation, as shown in Figure 4 below. In fact Figure 4 can be recolored in three colors to give it color symmetry, which means that every symmetry (rotation, reflection, etc.) of the uncolored pattern exactly permutes the colors of the fish in the colored pattern [Du2].

Figure 4: A pattern of angular fish based on the {6,6} tessellation.

Escher quite successfully overcame all of his criticisms of Circle Limit I in his print Circle Limit III, which has 4-color 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.

Figure 5: A computer rendition of Escher's Circle Limit III pattern, showing the underlying {8,3} tessellation.

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:

The value of A is about 79.97 degrees, which Escher accurately constructed to high precision.

One can imagine a three-parameter 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.

Figure 6: The pattern of fish on the 2003 Math Awareness Month poster.

It is necessary to use six colors to color the lines of fish symmetrically and adhere to the map-coloring 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 value of A is about 78.07 degrees.

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 computer-generated 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].

References

[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 0-8109-0858-1

[Co1] Coxeter, H. S. M., "Crystal symmetry and its generalizations," Royal Society of Canada(3), 51 (1957), 1-13.

[Co2] Coxeter, H. S. M., "The non-Euclidean symmetry of Escher's Picture 'Circle Limit III'," Leonardo, 12 (1979), 19-25, 32.

[Co3] Coxeter, H. S. M., "The Trigonometry of Escher's Woodcut 'Circle Limit III'," The Mathematical Intelligencer, 18 no. 4 (1996) 42-46. 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 0-534-98015-5
Republished as: Mathematical Recreations: A Collection in Honor of Martin Gardner, David A. Klarner, editor, Dover Publishers, 1998. ISBN 0-486-40089-1

[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. 1-2, 139-153.

[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. 286-296.

[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] Goodman-Strauss, Chaim, "Compass and straightedge in the Poincaré disk," Amer. Math. Monthly, 108 (2001), no. 1, 38-49.

[Gr1] Greenberg, Marvin, Euclidean and Non-Euclidean Geometries, 3rd Edition, W. H. Freeman and Co., 1993. ISBN 0-7167-2446-4

[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, 28-30 July 2000. ISBN 0-9665201-2-2 Web link: Abstract and PDF

[Ma1] Magnus, Wilhelm, Noneuclidean Tesselations and Their Groups, Academic Press, 1974. ISBN 0-12-465450-9

[Sc1] Schattschneider, Doris, Visions of Symmetry: Notebooks, Periodic Drawings, and Related Work of M. C. Escher, W. H. Freeman, New York, 1990. ISBN 0-7167-2126-0

[Sc2] Schattschneider, Doris, and Michele Emmer, editors, M. C. Escher's Legacy: A Centennial Celebration, Springer Verlag, 2003. ISBN 3-540-42458-X