Abstract
This article completes the classification of finite universal locally projective regular abstract polytopes, by summarising (with careful references) previously published results on the topic, and resolving the few cases that do not appear in the literature. In rank 4, all quotients of the locally projective polytopes are also noted. In addition, the article almost completes the classification of the infinite universal locally projective polytopes, except for the {{5,3,3,},{3,3,5}15} and its dual. It is shown that this polytope cannot be finite, but its existence is not established. The most remarkable feature of the classification is that a nondegenerate universal locally projective polytope \( \mathcal{P} \) is infinite if and only if the rank of \( \mathcal{P} \) is 5 and the facets of \( \mathcal{P} \) or its dual are the hemi-120-cell {5,3,3}15.
This is a preview of subscription content, access via your institution.
References
- [1]
H. S. M. Coxeter: Regular Polytopes, Methuen and Co., 1968.
- [2]
H. S. M. Coxeter: Ten Toroids and Fifty-Seven Hemi-dodecahedra, Geom. Dedicata 13 (1982), 87–99.
- [3]
H. S. M. Coxeter: A Symmetrical Arrangement of Eleven Hemi-icosahedra, Ann. Disc. Math. 20 (1984), 103–114.
- [4]
The GAP Group: GAP — Groups, Algorithms, and Programming, Version 4.4; http://www.gap-system.org (2005).
- [5]
B. Grünbaum: Regularity of Graphs, Complexes and Designs, in: Problèmes Combinatoires et Théorie des Graphes, Colloquium International CNRS, Orsay, 260, (1977), 191–197.
- [6]
M. I. Hartley: Polytopes of Finite Type, Discrete Math. 218 (2000), 97–108.
- [7]
M. I. Hartley: Quotients of Some Finite Universal Locally Projective Polytopes, Discrete Comput. Geom. 29 (2003), 435–443.
- [8]
M. I. Hartley: Locally Projective Polytopes of Type {4,3,...,3,p}, Journal of Algebra 290 (2005), 322–336.
- [9]
M. I. Hartley: Simpler Tests for Semisparse Subgroups, Annals of Combinatorics 10 (2006), 343–352.
- [10]
M. I. Hartley and D. Leemans: Quotients of a Locally Projective Polytope of Type {5,3,5}, Math. Z. 247 (2004), 663–674.
- [11]
M. I. Hartley and D. Leemans: Quotients of a Locally Projective Polytope of Type {5,3,5}: Errata; Math. Z. 253 (2006), 433–434.
- [12]
M. I. Hartley and D. Leemans: A New Petrie-like Construction for Abstract Polytopes, J. Comb. Theory A, to appear (2008). http://dx.doi.org/10.1016/j.jcta.2007.11.008
- [13]
P. McMullen: Locally Projective Regular Polytopes, J. Comb. Theory A 65 (1994), 1–10.
- [14]
P. McMullen and E. Schulte: Abstract Regular Polytopes, Cambridge University Press, 2002.
- [15]
J. Rotman: An Introduction to the Theory of Groups, Springer-Verlag, 1995.
- [16]
L. Schläfli: Theorie der Vielfachen Kontinuität, Denkschriften der Schweizerlichen Naturforschenden Gesellschaft 38 (1901), 1–237.
- [17]
E. Schulte: Amalgamation Of Regular Incidence-Polytopes, Proc. London Math. Soc. 56 (1988), 303–328.
- [18]
E. Schulte: Classification of Locally Toroidal Regular Polytopes, in: Polytopes: Abstract, Convex and Computational (T. Bisztriczky et al., eds.), 125–154, Kluwer, 1994.
Author information
Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hartley, M.I. An exploration of locally projective polytopes. Combinatorica 28, 299–314 (2008). https://doi.org/10.1007/s00493-008-2230-3
Received:
Published:
Issue Date:
Mathematics Subject Classification (2000)
- 51M20
- 52B15