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Realizations of Regular Toroidal Maps

Published online by Cambridge University Press:  20 November 2018

B. Monson  and
A. Ivić Weiss
B. Monson
Affiliation:
University of New Brunswick, Fredericton, New Brunswick, E3B 5A3
A. Ivić Weiss
Affiliation:
York University, Toronto, Ontario, M3J 1P3
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Abstract

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We determine and completely describe all pure realizations of the finite regular toroidal polyhedra of types {3, 6} and {6, 3}.

Keywords

51M2020F55regular mapsrealizations of polytopes
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 6 , 01 December 1999 , pp. 1240 - 1257
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Burgiel, H. and Stanton, D., Realizations of Regular Abstract Polyhedra of Types ﹛3, 6﹜ and ﹛6, 3﹜. Discrete and Comput. Geom., to appear.Google Scholar
[2] Coxeter, H. S. M., Regular skew polyhedra in three and four dimensions, and their topological analogues. Proc. London Math. Soc. 43 (1937), 33–62; reprinted in H. S. M. Coxeter, Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968, 75–105.Google Scholar
[3] Coxeter, H. S. M., Groups Generated by Unitary Reflections of Period Two. Canad. J. Math. 9 (1957), 243–272.Google Scholar
[4] Coxeter, H. S. M., Regular Complex Polytopes, 2nd Ed.. Cambridge Univ. Press, Cambridge (1991).Google Scholar
[5] Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups. Springer, New York, 1972.Google Scholar
[6] Grünbaum, B., Regularity of Graphs, Complexes and Designs. ColloquesCNRS, Problèmes Combinatoires et Theorie des Graphes, CNRS, Orsay, 1976, 191–197.Google Scholar
[7] Grünbaum, B., Regular polyhedra—old and new. Aequationes Math. 16(1977), 1–20.Google Scholar
[8] McMullen, P., Combinatorially regular polytopes. Mathematika 14(1967), 142–150.Google Scholar
[9] McMullen, P., Realizations of Regular Polytopes. AequationesMath. 37(1989), 38–56.Google Scholar
[10] McMullen, P., Modern Developments in Regular Polytopes. In: Polytopes: Abstract, Convex and Computational, (eds., Bisztriczky, T. et al.), Proc. NATO Advanced Study Institute, NATO ASI Series C 440, Kluwer, Dordrecht, 1994, 97–124.Google Scholar
[11] McMullen, P. and Schulte, E., Constructions for Regular Polytopes. J. Combin. Theory Ser. A 53(1990), 1–28.Google Scholar
[12] McMullen, P. and Schulte, E., Hermitian Forms and Locally Toroidal Regular Polytopes. Adv. inMath. 82(1990), 88–125.Google Scholar
[13] McMullen, P. and Schulte, E., Abstract Regular Polytopes. manuscript in preparation.Google Scholar
[14] Monson, B. and IvićWeiss, A., Realizations of regular toroidal maps of type {4, 4}. in Discrete and Comput. Geom., to appear.Google Scholar
[15] Vinberg, E. B., Linear Representations of Groups, Birkhäuser Verlag, Basel, 1989.Google Scholar