Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year


IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 18, Issue 4

Issues

Regular polyhedra in the 3-torus

Dedicated to Enrique Rodriguez Castillo, a beloved friend and colleague whom we all are going to miss

Antonio Montero
  • Corresponding author
  • Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0017

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.

Keywords: Regular polyhedra; 3-dimensional torus

MSC 2010: 52B15; 52B10; 51M20

  • [1]

    J. L. Arocha, J. Bracho, L. Montejano, Regular projective polyhedra with planar faces. I. Aequationes Math. 59 (2000), 55–73. MR1741470 Zbl 0952.52011Google Scholar

  • [2]

    J. Bracho, Regular projective polyhedra with planar faces. II. Aequationes Math. 59 (2000), 160–176. MR1741478 Zbl 0952.52012Google 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 63.0584.03Google Scholar

  • [4]

    H. S. M. Coxeter, Regular polytopes. Dover Publications, New York 1973. MR0370327Google Scholar

  • [5]

    H. S. M. Coxeter, W. O. J. Moser, Generators and relations for discrete groups. Springer 1972. MR0349820 Zbl 0239.20040Google 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 0506.51010Google 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 0588.51022Google Scholar

  • [8]

    B. Grünbaum, Regular polyhedra—old and new. Aequationes Math. 16 (1977), 1–20. MR0467497 Zbl 0381.51012Google Scholar

  • [9]

    B. Grünbaum, Uniform tilings of 3-space. Geombinatorics 4 (1994), 49–56. MR1294696 Zbl 0844.52022Google 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 1263.51016Google Scholar

  • [11]

    𝒫. McMullen, Regular polytopes of full rank. Discrete Comput. Geom. 32 (2004), 1–35. MR2060815 Zbl 1059.52019Google Scholar

  • [12]

    𝒫. McMullen, Four-dimensional regular polyhedra. Discrete Comput. Geom. 38 (2007), 355–387. MR2343312 Zbl 1134.52017Google Scholar

  • [13]

    𝒫. McMullen, Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom. 42 (2009), 224–260. MR2519878 Zbl 1178.52009Google Scholar

  • [14]

    𝒫. McMullen, E. Schulte, Regular polytopes in ordinary space. Discrete Comput. Geom. 17 (1997), 449–478. MR1455693 Zbl 0876.52003Google Scholar

  • [15]

    𝒫. McMullen, E. Schulte, Abstract regular polytopes, volume 92 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2002. MR1965665 Zbl 1039.52011Google Scholar

  • [16]

    D. Pellicer, A. Ivić Weiss, Combinatorial structure of Schulte’s chiral polyhedra. Discrete Comput. Geom. 44 (2010), 167–194. MR2639823 Zbl 1198.51010Google Scholar

  • [17]

    J. G. Ratcliffe, Foundations of hyperbolic manifolds. Springer 2006. MR2249478 Zbl 1106.51009Google Scholar

  • [18]

    E. Schulte, Chiral polyhedra in ordinary space. I. Discrete Comput. Geom. 32 (2004), 55–99. MR2060817 Zbl 1059.52020Google Scholar

  • [19]

    E. Schulte, Chiral polyhedra in ordinary space. II. Discrete Comput. Geom. 34 (2005), 181–229. MR2155719 Zbl 1090.52009Google Scholar

  • [20]

    𝒫. B. Yale, Geometry and symmetry. Dover Publications, New York 1988. MR1017247 Zbl 0701.51001Google Scholar

About the article


Received: 2016-05-03

Revised: 2016-08-02

Published Online: 2018-10-25

Published in Print: 2018-10-25


Citation Information: Advances in Geometry, Volume 18, Issue 4, Pages 431–450, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0017.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in