Abstract
We discuss the classification of rank 3 lattices preserved by finite orthogonal groups and derive from it the classification of regular polyhedra in the 3-dimensional torus. This classification is closely related to the classification of regular polyhedra in 3-space.
[1] J. L. Arocha, J. Bracho, L. Montejano, Regular projective polyhedra with planar faces. I. Aequationes Math. 59 (2000), 55–73. MR1741470 Zbl 0952.5201110.1007/PL00000128Search in Google Scholar
[2] J. Bracho, Regular projective polyhedra with planar faces. II. Aequationes Math. 59 (2000), 160–176. MR1741478 Zbl 0952.5201210.1007/PL00000122Search in Google Scholar
[3] H. S. M. Coxeter, Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues. Proc. London Math. Soc. (2) 43 (1937), 33–62. MR1575418 Zbl 0016.27101 JFM 63.0584.0310.1112/plms/s2-43.1.33Search in Google Scholar
[4] H. S. M. Coxeter, Regular polytopes. Dover Publications, New York 1973. MR0370327Search in Google Scholar
[5] H. S. M. Coxeter, W. O. J. Moser, Generators and relations for discrete groups. Springer 1972. MR0349820 Zbl 0239.2004010.1007/978-3-662-21946-1Search in Google Scholar
[6] A. W. M. Dress, A combinatorial theory of Grünbaum’s new regular polyhedra. I. Grünbaum’s new regular polyhedra and their automorphism group, Aequationes Math. 23 (1981), 252–265. MR689040 Zbl 0506.5101010.1007/BF02188039Search in Google Scholar
[7] A. W. M. Dress, A combinatorial theory of Grünbaum’s new regular polyhedra. II. Complete enumeration. Aequationes Math. 29 (1985), 222–243. MR819312 Zbl 0588.5102210.1007/BF02189831Search in Google Scholar
[8] B. Grünbaum, Regular polyhedra—old and new. Aequationes Math. 16 (1977), 1–20. MR0467497 Zbl 0381.5101210.1007/BF01836414Search in Google Scholar
[9] B. Grünbaum, Uniform tilings of 3-space. Geombinatorics 4 (1994), 49–56. MR1294696 Zbl 0844.52022Search in Google Scholar
[10] I. Hubard, A. Orbanić, D. Pellicer, A. I. Weiss, Symmetries of equivelar 4-toroids. Discrete Comput. Geom. 48 (2012), 1110–1136. MR3000577 Zbl 1263.5101610.1007/s00454-012-9444-2Search in Google Scholar
[11] 𝒫. McMullen, Regular polytopes of full rank. Discrete Comput. Geom. 32 (2004), 1–35. MR2060815 Zbl 1059.5201910.1007/s00454-004-0848-5Search in Google Scholar
[12] 𝒫. McMullen, Four-dimensional regular polyhedra. Discrete Comput. Geom. 38 (2007), 355–387. MR2343312 Zbl 1134.5201710.1007/s00454-007-1342-7Search in Google Scholar
[13] 𝒫. McMullen, Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom. 42 (2009), 224–260. MR2519878 Zbl 1178.5200910.1007/s00454-009-9186-ySearch in Google Scholar
[14] 𝒫. McMullen, E. Schulte, Regular polytopes in ordinary space. Discrete Comput. Geom. 17 (1997), 449–478. MR1455693 Zbl 0876.5200310.1007/PL00009304Search in Google Scholar
[15] 𝒫. McMullen, E. Schulte, Abstract regular polytopes, volume 92 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2002. MR1965665 Zbl 1039.5201110.1017/CBO9780511546686Search in Google Scholar
[16] D. Pellicer, A. Ivić Weiss, Combinatorial structure of Schulte’s chiral polyhedra. Discrete Comput. Geom. 44 (2010), 167–194. MR2639823 Zbl 1198.5101010.1007/s00454-010-9247-2Search in Google Scholar
[17] J. G. Ratcliffe, Foundations of hyperbolic manifolds. Springer 2006. MR2249478 Zbl 1106.51009Search in Google Scholar
[18] E. Schulte, Chiral polyhedra in ordinary space. I. Discrete Comput. Geom. 32 (2004), 55–99. MR2060817 Zbl 1059.5202010.1007/s00454-004-0843-xSearch in Google Scholar
[19] E. Schulte, Chiral polyhedra in ordinary space. II. Discrete Comput. Geom. 34 (2005), 181–229. MR2155719 Zbl 1090.5200910.1007/s00454-005-1176-0Search in Google Scholar
[20] 𝒫. B. Yale, Geometry and symmetry. Dover Publications, New York 1988. MR1017247 Zbl 0701.51001Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds
Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds