In this paper it is shown that any (abstract) polytope P is a quotient of a regular polytope M by some subgroup N of the automorphism group W of M, and also that isomorphic polytopes are quotients of M by conjugate subgroups of W . This extends work published in 1980 by Kato, who proved these results for a restricted class of polytopes which he called "regular". The methods used here are more elementary, and treat the problem in a purely nongeometric manner.
This article announces the creation of an atlas of small regular abstract polytopes. The atlas contains information about all regular abstract polytopes whose automorphism group has order 2000 or less, except those of order 512k where k ≥ 1. The article explains also the techniques used to create the atlas, and gives some summary tables. At the time of printing, the url for the atlas is http://www.abstract-polytopes.com/atlas.
We construct chiral abstract polytopes in two different ways. Firstly we seek them as quotients of regular polytopes arising from the Atlas of Small Regular Polytopes (http: //www.abstract-polytopes.com/atlas/); the resulting atlas of chiral polytopes atlas is available on the website http://www.abstract-polytopes.com/ chiral/. Secondly, for each almost simple group Γ such that S ≤ Γ ≤ Aut(S) where S is a simple group and Γ is a group of order less than 900,000 listed in the Atlas of Finite Groups, we give, up to isomorphism, the number of abstract chiral polytopes on which Γ acts regularly. The results have been obtained using a series of MAGMA programs. All these polytopes are made available on the third author's website, at http: //math.auckland.ac.nz/˜dleemans/CHIRAL/.
This article completes the classification of finite universal locally projective regular abstract polytopes, by summarising (with careful references) previously published results on the topic, and resolving the few cases that do not appear in the literature. In rank 4, all quotients of the locally projective polytopes are also noted. In addition, the article almost completes the classification of the infinite universal locally projective polytopes, except for the {{5, 3, 3}, {3, 3, 5}15 } and its dual. It is shown that this polytope cannot be finite, but its existence is not established. The most remarkable feature of the classification is that a nondegenerate universal locally projective polytope P is infinite if and only if the rank of P is 5 and the facets of P or its dual are the hemi-120-cell {5, 3, 3}15.
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