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Polytopes with an Axis Of Symmetry

Published online by Cambridge University Press:  20 November 2018

P. McMullen  and
G. C. Shephard
P. McMullen
Affiliation:
Michigan State University, East Lansing, Michigan
G. C. Shephard
Affiliation:
Michigan State University, East Lansing, Michigan
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During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 265 - 287
Copyright
Copyright © Canadian Mathematical Society 1970

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