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, E2, S2, and H2, by bounded regular tiles. For instance, there exist infinitely many regular tessellations of the hyperbolic plane H2 by curved hyperbolic equilateral triangles whose vertex angles are 2π/d for d=7,8,9,… On the other hand, we prove that there is no curved hyperbolic regular tetrahedron which tessellates the three-dimensional hyperbolic space H3. We also show that a regular tessellation of H3 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 H4. If n>4, then there exists no regular tessellation of Hn.