An exploration of locally projective polytopes

Abstract

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 \( \mathcal{P} \) is infinite if and only if the rank of \( \mathcal{P} \) is 5 and the facets of \( \mathcal{P} \) or its dual are the hemi-120-cell {5,3,3}15.

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Correspondence to Michael I. Hartley.

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Hartley, M.I. An exploration of locally projective polytopes. Combinatorica 28, 299–314 (2008). https://doi.org/10.1007/s00493-008-2230-3

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Mathematics Subject Classification (2000)

  • 51M20
  • 52B15