Discrete & Computational Geometry

, Volume 42, Issue 2, pp 224–260

Regular Apeirotopes of Dimension and Rank 4

Article

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 R0,…,dim R3) of the mirrors R0,…,R3 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.

Keywords

Polytope Abstract Regular Realization Apeirotope Rank Dimension 

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References

  1. 1.
    Arocha, J.L., Bracho, J., Montejano, L.: Regular projective polyhedra with planar faces, I. Aequ. Math. 59, 55–73 (2000) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bracho, J.: Regular projective polyhedra with planar faces, II. Aequ. Math. 59, 160–176 (2000) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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) MATHCrossRefGoogle Scholar
  4. 4.
    Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980) Google Scholar
  5. 5.
    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) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequ. Math. 29, 222–243 (1985) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frucht, R., Graver, J.E., Watkins, M.E.: The groups of the generalized Petersen graphs. Proc. Camb. Philos. Soc. 70, 211–218 (1971) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grünbaum, B.: Regular polyhedra—old and new. Aequ. Math. 16, 1–20 (1977) MATHCrossRefGoogle Scholar
  9. 9.
    McMullen, P.: Realizations of regular polytopes. Aequ. Math. 37, 38–56 (1989) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    McMullen, P.: The regular polyhedra of type {p,3} with 2p vertices. Geom. Dedic. 43, 285–289 (1992) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    McMullen, P.: Realizations of regular apeirotopes. Aequ. Math. 47, 223–239 (1994) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    McMullen, P.: The groups of the regular star-polytopes. Can. J. Math. 50(2), 426–448 (1998) MATHMathSciNetGoogle Scholar
  13. 13.
    McMullen, P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    McMullen, P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    McMullen, P.: Regular polytopes of nearly full rank (2009, in preparation) Google Scholar
  16. 16.
    McMullen, P., Monson, B.R.: Realizations of regular polytopes, II. Aequ. Math. 65, 102–112 (2003) MATHMathSciNetGoogle Scholar
  17. 17.
    McMullen, P., Schulte, E.: Constructions for regular polytopes. J. Comb. Theory, Ser. A 53, 1–28 (1990) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    McMullen, P., Schulte, E.: Regular polytopes from twisted Coxeter groups and unitary reflexion groups. Adv. Math. 82, 35–87 (1990) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University College LondonLondonUK

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