Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds

We first briefly summarize several well-known properties of regular tessellations of the three two-dimensional maximally symmetric manifolds, E-2, S-2, and H-2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H-2 by curved hyperbolic equilateral triangles whose vertex angles are 2 pi/d for d=7,8,9, horizontal ellipsis On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H-3. We also show that a regular tessellation of H-3 can only consist of the hyperbolic cubes, hyperbolic regular icosahedra, or two types of hyperbolic regular dodecahedra. There exist only two regular hyperbolic space-fillers of H-4. If n>4, then there exists no regular tessellation of H-n.