A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration |
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Authors: | Andreas W M Dress |
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Institution: | (1) Universität Bielefeld, Fakultät für Mathematik, D-4800 Bielefeld, West Germany |
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Abstract: | The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. 5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. 3], 2] or 1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in 4], it turns out that the list of examples given in 5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE
3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE
3, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE
n
is worked out in §9. |
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Keywords: | Primary 51M20 51F15 |
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