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- Title
Regular Tessellations of Maximally Symmetric Hyperbolic Manifolds.
- Authors
Brandts, Jan; Křížek, Michal; Somer, Lawrence
- Abstract
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 π / 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 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 .
- Subjects
HYPERBOLIC geometry; HYPERBOLIC spaces; TESSELLATIONS (Mathematics); ICOSAHEDRA; ANGLES; TETRAHEDRA; TRIANGLES
- Publication
Symmetry (20738994), 2024, Vol 16, Issue 2, p141
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym16020141