Discrete & Computational Geometry

, Volume 42, Issue 2, pp 224–260

Regular Apeirotopes of Dimension and Rank 4


DOI: 10.1007/s00454-009-9186-y

Cite this article as:
McMullen, P. Discrete Comput Geom (2009) 42: 224. doi:10.1007/s00454-009-9186-y


In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in \(\mathbb{E}^{4}\) thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim R0,…,dim R3) of the mirrors R0,…,R3 of the generating reflexions of the symmetry groups plays a rôle, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.



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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University College LondonLondonUK