Douglas Dunham

Creating repeating hyperbolic patterns

A process for creating repeating patterns of the hyperbolic plane is described. Unlike the Euclidean plane, the hyperbolic plane has infinitely many different kinds of repeating patterns. The Poincare circle model of hyperbolic geometry has been used by the artist M. C. Escher to display interlocking, repeating, hyperbolic patterns. A program has been designed which will do this automatically. The user enters a motif, or basic subpattern, which could theoretically be replicated to fill the hyperbolic plane. In practice, the replication process can be iterated sufficiently often to appear to fill the circle model. There is an interactive “boundary procedure” which allows the user to design a motif Which will be replicated into a completely interlocking pattern. Duplication of two of Eschers patterns and some entirely new patterns are included in the paper.

Infinite Hamiltonian paths in Cayley digraphs of hyperbolic symmetry groups

Abstract The hyperbolic symmetry groups [ p,q ], [ p,q ] + , and [ p + , q ] have certain natural generating sets. We determine whether or not the corresponding Cayley digraphs have one-way infinite or two-way infinite directed Hamiltonian paths. In addition, the analogous Cayley graphs are shown to have both one-way infinite and two-way infinite Hamiltonian paths.

M.C. Escher’s Use of the Poincaré Models of Hyperbolic Geometry

The artist M.C. Escher was the first artist to create patterns in the hyperbolic plane. He used both the Poincare disk model and the Poincare half-plane model of hyperbolic geometry. We discuss some of the theory of hyperbolic patterns and show Escher-inspired designs in both of these models.

A Property of Area and Perimeter

We describe an algorithm that creates a fractal pattern within a planar region R by ranplacing within it progressively smaller copies of a subpattern or fill-shape. After placing i copies of the fill-shape, we use the term gasket to describe the unfilled part of R. The size of the next fill-shape is determined by a constant \(\gamma \) times the remaining area of the gasket divided by the total perimeter of the gasket. Experimental evidence is presented indicating that the areas of the fill-shapes obey an inverse power law for large i.