Use of Models of Hyperbolic Geometry in the Creation of Hyperbolic Patterns

In 1958, the Dutch artist M.C. Escher became the first person to create artistic patterns in hyperbolic geometry. He used the Poincar é circle model of hyperbolic geometry. Slightly more than 20 years later, my students and I implemented a computer program that could draw repeating hyperbolic patterns in this model. The program made substantial use of the Weierstrass model of hyperbolic geometry as an intermediate step. We have also made use of the Klein model of hyperbolic geometry, both for approximating hyperbolic lines and for transforming motifs from one set of combinatorial values to another.