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Regular 4-polytopes related to general orthogonal groups

Part of: Real and complex geometry

Published online by Cambridge University Press:  26 February 2010

B. Monson  and
Asia Ivić Weiss
B. Monson
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada, E3B 5A3.
Asia Ivić Weiss
Affiliation:
Department of Mathematics, York University, 4700 Keele St., North York, Ontario, Canada, M3J 1P3.
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Abstract

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p}. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.

MSC classification

Secondary: 51M20: Polyhedra and polytopes; regular figures, division of spaces

Information

Type
Research Article
Information
Mathematika , Volume 37 , Issue 1 , June 1990 , pp. 106 - 118
Copyright
Copyright © University College London 1990

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References

1.Artin, E.. Geometric Algebra (Interscience, New York, 1957).Google Scholar
2.Bianchi, L.. Geometrische Darstellung der Gruppen Iinearer Substitutionen mit ganzen complexen Coefficienten nebst Anwendungen auf die Zahlentheorie. Math. Ann., 38 (1891), 313–333.CrossRefGoogle Scholar
3.Bourbaki, N.. Groupes et algebres de Lie, chap, iv-vi (Hermann, Paris, 1968).Google Scholar
4.Conway, J. H. et al. Atlas of Finite Groups (Clarendon Press, Oxford, 1985).Google Scholar
5.Coxeter, H. S. M.. Abstract Definitions for the Symmetry Groups of the Regular Polytopes, in Terms of Two Generators. Part Two: the Rotation Groups. Proa Cambridge Phil. Soc, 33 (1937), 315–324.CrossRefGoogle Scholar
6.Coxeter, H. S. M.. Twelve Geometric Essays (Southern Illinois University Press, Carbondale, 1968).Google Scholar
7.Coxeter, H. S. M.. Regular Polytopes (Dover, New York, 1973).Google Scholar
8.Danzer, L. and Schulte, E.. Regulare Inzidenzkomplexe I. Geom. Dedicata, 13 (1982), 295–308.CrossRefGoogle Scholar
9.Dickson, L. E.. Linear Groups, with an exposition of the Galois Field theory (Dover, New York, 1958).Google Scholar
10.Homographies, P. Du Val., Quaternions and Rotations (Clarendon Press, Oxford, 1964).Google Scholar
11.McMullen, P.. Realizations of Regular Polytopes. Aequationes Math., 37 (1989), 38–56.CrossRefGoogle Scholar
12.McMullen, P.. Regular Polyhedra Related to Projective Linear Groups. To appear in Discrete Math.Google Scholar
13.McMullen, P. and Schulte, E.. Constructions for Regular Polytopes. J. Comb. Theory (A). To appear.Google Scholar
14.McMullen, P. and Schulte, E.. Hermitian Forms and Locally Toroidal Regular Polytopes. To appear.Google Scholar
15.McMullen, P. and Schulte, E.. Regular Polytopes from Twisted Coxeter Groups. Math. Zeit., 201 (1989), 209–226.CrossRefGoogle Scholar
16.McMullen, P. and Schulte, E.. Regular Polytopes from Twisted Coxeter Groups and Unitary Reflection Groups. Advances in Mathematics. In press.Google Scholar
17.McMullen, P. and Schulte, E.. Regular Polytopes of Type {4, 4, 3} and {4, 4, 4}. To appear in Combinatorica.Google Scholar
18.McMullen, P. and Schulte, E.. Abstract Regular Polytopes. Manuscript in preparation.Google Scholar
19.Schulte, E.. Regulare Inzidenzkomplexe II. Geom. Dedicata, 14 (1983), 35–56.Google Scholar
20.Schulte, E.. Regular Incidence-Polytopes with Euclidean or Toroidal Faces and Vertex-Figures. J. Comb. Theory (A), 40 (1985), 305–330.CrossRefGoogle Scholar
21.Schulte, E.. Amalgamation of Regular Incidence-Polytopes. Proc. London Math. Soc, 56 (1988), 303–328.CrossRefGoogle Scholar
22.Sunday, J. G.. Presentations of the groups SL (2, m) and PSL (2, m). Canad. J. Math., 24 (1972), 1129–1131.CrossRefGoogle Scholar
23.Vinberg, E. B.. On Groups of Unit Elements of Certain Quadratic Forms. Math. USSR-Sb., 16 (1972), 17–35.Google Scholar
24.Weiss, A. I.. Incidence-Polytopes with Toroidal Cells. Discrete Comput. Geom., 4 (1989), 55–73.Google Scholar
25.Weiss, A. I.. Some Infinite Families of Finite Incidence-Polytopes. J. Comb. Theory (A). In press.Google Scholar
26.Wilker, J. B.. Inversive Geometry. The Geometric Vein—the Coxeter Festschrift (Springer, New York, 1981), 379–442.Google Scholar