Regular polytopes in ordinary space (Q1356086)

The authors give a new proof of the completeness of the enumeration of the regular apeirohedra (that is, the regular infinite polyhedra) of rank 3 in \(\mathbb E^3\). Previously the proof was obtained by \textit{A. W. M. Dress} in Aequationes Math. 23, 252-265 (1981; Zbl 0506.51010) and ibid. 29, 222-243 (1985; Zbl 0588.51022). Besides, the paper contains a complete classification of the discrete regular polytopes in \(\mathbb E^3\): the authors describe seven new discrete regular apeirotopes of rank 4 in the ordinary space, to add to the familiar honeycomb \(\{ 4,3,4\} \) of cubes, and prove that there are no others. Presentations of the symmetry groups are given of those discrete regular apeirohedra in \(\mathbb E^3\) whose groups are affinely irreducible. New pairings are introduced between the finite regular polyhedra. The notion of abstract polytope is used systematically.