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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

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
51F15 Reflection groups, reflection geometries
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References:

[1] Abels, H. andDress, A.,An algebraic version of a theorem of L. Bieberbach, concerning invariant subspaces of discrete isometry groups. Submitted to the J. Algebra.
[2] Bieberbach, L.,Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung). Math. Ann.70 (1910), 297–336. · doi:10.1007/BF01564500
[3] Brown, H., Bulow, R., Neubüser, J., Wonratschek, H., andZassenhaus, H.,Crystallographic groups of four-dimensional space. Wiley, New York, 1978.
[4] Dress, A.,A combinatorial theory of Grünbaum’s new regular polyhedra, Part I: Grünbaum’s new regular polyhedra and their automorphism group. Aequationes Math.23 (1981), 252–264. · Zbl 0506.51010 · doi:10.1007/BF02188039
[5] Grünbaum, B.,Regular polyhedra–old and new. Aequationes Math.16 (1977), 1–20. · Zbl 0381.51012 · doi:10.1007/BF01836414
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