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Quasi-Regular Polytopes of Full Rank

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Abstract

A polytope \({{ \mathsf {P}}}\) in some euclidean space is called quasi-regular if each facet \({{ \mathsf {F}}}\) of \({{ \mathsf {P}}}\) is regular and the symmetry group \({\mathbf {G}}({{ \mathsf {F}}})\) of \({{ \mathsf {F}}}\) is a subgroup of the symmetry group \({\mathbf {G}}({{ \mathsf {P}}})\) of \({{ \mathsf {P}}}\). Further, \({{ \mathsf {P}}}\) is of full rank if its rank and dimension are the same. In this paper, the quasi-regular polytopes of full rank that are not regular are classified. Similarly, an apeirotope of full rank sits in a space of one fewer dimension; the discrete quasi-regular apeirotopes that are not regular are also classified here. One curiosity of the classification is the difference between even and odd dimensions, in that certain families are present in \({\mathbb {E}}^d\) if d is even, but are absent if d is odd.

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Acknowledgements

We wish to thank the three referees, whose useful suggestions have led to substantial improvements in the paper. We are particularly grateful to the one who pointed out numerous mistakes, in particular a flaw in the proof of Theorem 6.2, and that it did not hold in rank 4 (which resulted in examples in \({\mathbb {E}}^3\) and \({\mathbb {E}}^4\) that were previously missed), and also drew our attention to a planar apeirohedron that had been omitted (in fact, basically that illustrated in (9.3)), suggesting that there would be similar ones.

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Correspondence to Peter McMullen.

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McMullen, P. Quasi-Regular Polytopes of Full Rank. Discrete Comput Geom 66, 475–509 (2021). https://doi.org/10.1007/s00454-021-00304-5

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