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Polytopes with an Axis Of Symmetry

  • P. McMullen (a1) and G. C. Shephard (a1)

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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 PEd 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.

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References

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1. Cartan, E., Complement au mémoire sur la géométrie des groupes simples, Ann. Mat. Pura Appl. (4) 5 (1928), 253260.
2. Grùnbaum, B., Convex polytopes (Wiley, London-New York-Sydney, 1967).
3. McMullen, P. and Shephard, G. C., Diagrams for centrally symmetric polytopes, Mathematika 15 (1968), 123138.
4. Pedes, M. A. and Shephard, G. C., Facets and nonfacets of convex polytopes, Acta Math. 119 (1967), 113145.
5. Pölya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen una chemische Verbindungen, Acta Math. 68 (1937), 145254.
6. Radon, J., Mengen konvexer Kbrper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921), 113115.
7. Schur, I., Ueber Gruppen periodischer Substitutionen, S.-B. Preuss. Akad. Wiss. 1911, 619627.
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Polytopes with an Axis Of Symmetry

  • P. McMullen (a1) and G. C. Shephard (a1)

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