“Creating repeating hyperbolic patterns” by Dunham, Lindgren and Witte
Conference:
Type(s):
Title:
- Creating repeating hyperbolic patterns
Presenter(s)/Author(s):
Abstract:
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 Escher’s patterns and some entirely new patterns are included in the paper.
References:
1. Alexander, H. Periodic Designs in the Euclidean and Hyperbolic Planes, Realized by Means of Computer plus Plotter, 1978 (unpublished).
2. Coxeter, H.S.M. Crystal symmetry and its generalizations. Trans. Royal Soc. Canada (3), 51(1957), 1-13.
3. Coxeter, H. S.M. Introduction to Geometry, Wiley, New York, 1961, (2nd ed. 1969)
4. Coxeter, H. S.M. and Moser, W.O.J. Generators and Relations for Discrete Groups, Springer-Verlag, New York, 1957 (4th ed. 1980)
5. Fricke, R. and Klein, F. Vorlesungen uber die Theorie der elliptischen Modulfunktionen, (Publisher unknown), Leipzig, 1890.
6. Locher, J. L. (Editor) The World of M. C. Escher, Abrams, New York, 1971.
