Discrete & Computational Geometry

, Volume 38, Issue 2, pp 355–387

Four-Dimensional Regular Polyhedra


DOI: 10.1007/s00454-007-1342-7

Cite this article as:
McMullen, P. Discrete Comput Geom (2007) 38: 355. doi:10.1007/s00454-007-1342-7


This paper completes the classification of the four-dimensional (finite) regular polyhedra, of which those with planar faces were—in effect—found by Arocha, Bracho and Montejano. However, the methods employed here are in the same spirit as those used in the description of all three-dimensional regular polytopes by this author and Schulte, and the regular polytopes of full rank by this author. The procedure has two stages. First, the possible dimension vectors (dim R0,dim R1,dim R2) of the mirrors R0,R1,R2 of the generating reflexions of the symmetry groups are determined. Second, all polyhedra with a given dimension vector are found. Most of the polyhedra are related to four-dimensional Coxeter groups, although one class has to be approached using quaternions.

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© Springer 2007

Authors and Affiliations

  1. 1.Department of Mathematics, University College London, Gower StreetLondon WC1E 6BTEngland