Artistic Patterns in Hyperbolic Geometry

Abstract

From antiquity, humans have created 2-dimensional art on flat surfaces (the Euclidean plane) and on surfaces of spheres. However, it wasn't until recently that they have created art in the third classical geometry, the hyperbolic plane. M. C. Escher was the first person to do so, doing all the needed constructions laboriously by hand. To exhibit the true hyperbolic nature of such art, the pattern must exhibit symmetry and repetition. Thus, it is natural to use a computer to avoid the tedious hand constructions performed by Escher. We show a number of hyperbolic patterns, which are created by combining mathematics, artistic considerations, and computer technology.

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