Automatic Generation of Hyperbolic Tilings

1We cover the fundamentals of plane hyperbolic geometry and describe a method to generate hyperbolic tilings by computer, based on the recently developed theory of automatic groups. Regular tilings of the Euclidean plane are a very common pictorial motif, mar-velously represented, for example, by the patterns on the walls of the Alhambra, in Spain, and by several of the works of M. C. Escher [1]. Many readers will also be familiar with Escher’s woodcut entitled Circle