Families of Escher Patterns

Although M.C. Escher is best known for his repeating Euclidean patterns of interlocking motifs, he also designed patterns for the sphere and hyperbolic plane. In some cases it is evident that he modified the motif of one pattern to obtain a new pattern with different parameters, different color symmetry, or even a different geometry. For example, Escher transformed his planar “angels and devils” pattern (Fig. 1) onto the sphere and the hyperbolic plane (Fig. 2), thus making use of each of the three classical geometries. H.S.M. Coxeter gives an interesting discussion of these three patterns and their symmetry groups [3, pp. 197–209]; a picture of the sphere is in [8, p. 92]. Figs. 1 and 2 are two examples from a doubly infinite family of angels and devils patterns parameterized by p, the number of figures whose feet meet at a point, and q, the number of devil wing tips meeting at a rotation center. The values of p and q are 4 and 4 for Fig. 1, and 6 and 4 for Fig. 2.