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 ℍ^𝑛.