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Chiral polytopes of full rank exist only in ranks 4 and 5

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Abstract

Skeletal chiral polytopes of rank n in a Euclidean space must have as ambient space \({\mathbb {E}}^d\) for some \(d \ge n\) if they are finite, or some \(d \ge n-1\) if they are infinite. If the dimension attains the lower bound just mentioned, we say that the polytope is of full rank. In this article it is proven that a chiral polytope of full rank can only have rank 4 or 5.

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Acknowledgements

The author was supported by PAPIIT-UNAM under project grant IN100518 and CONACYT “Fondo Sectorial de Investigación para la Educación” under grant A1-S-10839. The author also thanks Barry Monson for helpful discussion, as well as the anonymous referee for pointing out invalid arguments in previous versions, whose replacements substantially improved this article.

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  1. Centro de Ciencias Matemáticas, UNAM, Campus Morelia, C.P. 58190, Morelia, Michoacán, Mexico

    Daniel Pellicer

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  1. Daniel Pellicer
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Correspondence to Daniel Pellicer.

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Pellicer, D. Chiral polytopes of full rank exist only in ranks 4 and 5. Beitr Algebra Geom 62, 651–665 (2021). https://doi.org/10.1007/s13366-020-00545-0

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