, Volume 38, Issue 2, pp 355-387
Date: 11 Dec 2007

Four-Dimensional Regular Polyhedra

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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.