A combinatorial theory of Grünbaum’s new regular polyhedra. II: Complete enumeration. (English) Zbl 0588.51022

From the author’s abstract: ”The new regular polyhedra as defined by B. Grünbaum in 1977 [ibid 16, 1-20 (1977; Zbl 0381.51012)] are completely enumerated. By means of a theorem of Bieberbach concerning the existence of invariant affine subspaces for discrete affine isometry groups, the standard crystallographic restrictions are established for the isometry groups of the non-finite (Grünbaum-) polyhedra. Then, using an appropriate classification scheme which is suggested by group theoretical investigations, it turns out that the list of examples given in the original Grünbaum paper is essentially complete except for one additional polyhedron.”
[For part I of the author’s paper see ibid. 23, 252-265 (1981; Zbl 0506.51010)].
Reviewer: C.Garner


51M20 Polyhedra and polytopes; regular figures, division of spaces
51F15 Reflection groups, reflection geometries
Full Text: DOI EuDML


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