Regular tessellations of maximally symmetric hyperbolic manifolds

0581986 - MÚ 2025 RIV CH eng J - Journal Article
Brandts, J. - Křížek, Michal - Somer, L.
Regular tessellations of maximally symmetric hyperbolic manifolds.
Symmetry-Basel. Roč. 16, č. 2 (2024), č. článku 141. E-ISSN 2073-8994
R&D Projects: GA ČR(CZ) GA24-10586S
Institutional support: RVO:67985840
Keywords : Euclidean space * spherical geometry * hyperbolic geometry * hypersphere
OECD category: Pure mathematics
Impact factor: 2.7, year: 2022
Method of publishing: Open access
https://doi.org/10.3390/sym16020141

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.
Permanent Link: https://hdl.handle.net/11104/0350121