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Two compounds of antiprisms in R4

Published online by Cambridge University Press:  24 October 2008

Anthony Smith
Affiliation:
Queen's University, Belfast

Extract

A polygon is said to be uniform if and only if it is connected and regular. An n-dimensional polytope (n > 2) is called uniform if it has uniform cells and possesses a symmetry group transitive on the vertices. A polytope is either connected or a compound. If it is a compound we also require the components to be isomorphic and to have no cells of dimension ≥ n − 1 in common. The connected 3-dimensional uniform polytopes have been investigated by Coxeter, Longuet-Higgins and Miller (1). The convex 4-dimensional polytopes have been investigated by Conway and Guy (2). Compound polytopes have been less extensively described than connected polytopes, but see, for example, (3), (4). Based on 2- and 3-dimensional experience, one might navely conjecture: ‘given a symmetry group , a uniform polytope P and an integer n, if a uniform compound polytope with symmetry group and consisting of n copies of P exists, then it is either unique or else a member of a one-parameter family’.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Coxeter, H. S. M., Longuet-Higgins, M. S. and Miller, J. C. P.Uniform polyhedra. Philos. Trans. Roy. Soc. London, Ser. A 246 (1954), 401450.Google Scholar
(2)Conway, J. H.Four-dimensional Archimedean polytopes. Proc. Coll. Convexity, Copenhagen, 1965 (1967), 3839.Google Scholar
(3)Brückner, M.Über die gleicheckig-gleichflächigen, diskontinuierlichen und nicht-konvexen Polyeder. Nova Acta (1906), Kaiserliche Leopoldinisch-Carolinische Deutsche Akademie der Naturforscher.Google Scholar
(4)Smith, A.Uniform compounds and the group A 4. Proc. Cambridge Philos. Soc. 75 (1974), 115117.CrossRefGoogle Scholar