Transformation of Hyperbolic Escher Patterns

Douglas Dunham

Computer scientist, mathematician (b. Pasadena, California, U.S.A., 1938)
Address: Department of Computer Science 320HH, University of Minnesota-Duluth, Duluth, Minnesota 55812-2496, U.S.A.; E-mail:; WWW home page:
Fields of interest: Hyperbolic geometry, M. C. Escher, computer graphics algorithms, computer generated art.
Publications: Families of Escher patterns, presented at M. C. Escher 1898-1998, An International Congress, June, 1998, Rome Italy; The Symmetry of hyperbolic Escher patterns, Proceedings of the Third Interdisciplinary Symmetry Congress (of the International Society for Interdisciplinary Studies of Symmetry: ISIS-Symmetry), August, 1995, Alexandria, VA; Artistic patterns with hyperbolic symmetry, Proceedings of Symmetry of Structure, An Interdisciplinary Symposium, August, 1989, Budapest, Hungary; Creating hyperbolic Escher patterns, M. C. Escher, Art and Science, H. S. M. Coxeter, et al., eds., North-Holland, New York, 1986, pp. 241-248; Hyperbolic symmetry, Computers & Mathematics with Applications, Vol. 12B, Nos. 1/2, 1986, pp. 139-153.


Abstract: The Dutch artist M. C. Escher is known for his repeating patterns of interlocking motifs. Most of Escher's patterns are Euclidean patterns, but he also designed some for the surface of the sphere and others for the hyperbolic plane, thus making use of all three classical geometries: Euclidean, spherical, and hyperbolic. In some cases it is evident that he applied a transformation to one of his patterns to obtain a new pattern, thus changing the symmetry of the original pattern, sometimes even forcing it onto a different geometry. In fact Escher transformed his Euclidean Pattern Number 45 of angels and devils both onto the sphere, Heaven and Hell on a carved maple sphere, and onto the hyperbolic plane, Circle Limit IV. A computer program has been written that converts one hyperbolic pattern to another by applying a transformation to its motif. We will describe the method used by this program.


  1. Introduction

  2. Hyperbolic Geometry

  3. Repeating Patterns and Regular Tessellations

  4. Transformation of a Pattern

  5. Examples Based on Circle Limits I and II

  6. Examples Based on Circle Limits III and IV

  7. Future Work, References, PostScript files