We study the set of bounded geodesies of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.

A polygon splitting algorithm is a combinatorial recipe. The description and the implementation of polygon splitting should not depend on the embedding geometry. Whether a polygon is being split in Euclidean, in spherical, in oriented projective, or in hyperbolic geometry should not be part of the description of the algorithm. The algorithm should be purely combinatorial, or geometry free. The ...

Chinese checkers is a game played on a hexagonal grid. This regular hexagonal tesselation is an artifact of Euclidean geometry that provides a fair playing eld only for games of two, three, four or six players. Hyperbolic geometry allows tessallations of the plane by regular polygons with any number of sides. Hyperbolic versions of the chinese checkers board permit fair games with ve, seven and...

Images are created using probabilistic iterated function systems that involve both affine transformations of the plane and isometries of hyperbolic geometry. Figures of attractors with striking hyperbolic symmetry are the result.

The artist M.C. Escher was the first artist to create patterns in the hyperbolic plane. He used both the Poincaré disk model and the Poincaré 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.