Abstract
In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in \(\mathbb{E}^{4}\) thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim R 0,…,dim R 3) of the mirrors R 0,…,R 3 of the generating reflexions of the symmetry groups plays a rôle, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.
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Arocha, J.L., Bracho, J., Montejano, L.: Regular projective polyhedra with planar faces, I. Aequ. Math. 59, 55–73 (2000)
Bracho, J.: Regular projective polyhedra with planar faces, II. Aequ. Math. 59, 160–176 (2000)
Coxeter, H.S.M.: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. London Math. Soc. (2) 43, 33–62 (1937). Reprinted with amendments in Twelve Geometric Essays, pp. 76–105. Southern Illinois University Press, Carbondale (1968)
Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, I: Grünbaum’s new regular polyhedra and their automorphism group. Aequ. Math. 23, 252–265 (1981)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequ. Math. 29, 222–243 (1985)
Frucht, R., Graver, J.E., Watkins, M.E.: The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971)
Grünbaum, B.: Regular polyhedra—old and new. Aequ. Math. 16, 1–20 (1977)
McMullen, P.: Realizations of regular polytopes. Aequ. Math. 37, 38–56 (1989)
McMullen, P.: The regular polyhedra of type {p,3} with 2p vertices. Geom. Dedic. 43, 285–289 (1992)
McMullen, P.: Realizations of regular apeirotopes. Aequ. Math. 47, 223–239 (1994)
McMullen, P.: The groups of the regular star-polytopes. Can. J. Math. 50(2), 426–448 (1998)
McMullen, P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004)
McMullen, P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007)
McMullen, P.: Regular polytopes of nearly full rank (2009, in preparation)
McMullen, P., Monson, B.R.: Realizations of regular polytopes, II. Aequ. Math. 65, 102–112 (2003)
McMullen, P., Schulte, E.: Constructions for regular polytopes. J. Comb. Theory, Ser. A 53, 1–28 (1990)
McMullen, P., Schulte, E.: Regular polytopes from twisted Coxeter groups and unitary reflexion groups. Adv. Math. 82, 35–87 (1990)
McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997)
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002)
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McMullen, P. Regular Apeirotopes of Dimension and Rank 4. Discrete Comput Geom 42, 224–260 (2009). https://doi.org/10.1007/s00454-009-9186-y
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DOI: https://doi.org/10.1007/s00454-009-9186-y