Chirality and projective linear groups
This paper treats (abstract) polytopes of rank \(n\) which are defined by the properties of the face lattice of ordinary \(n\)-dimensional convex polytopes. A polytope is called chiral if it is in a certain sense symmetrical by rotation but not by reflexion, thus giving rise to the phenomenon of enantiomorphism. Important examples of regular or chiral polytopes of rank 3 correspond to maps on the torus. They are derived from the regular tessellations \(\{4,4\}\), \(\{3,6\}\) or \(\{6,3\}\) of the plane. Here the authors construct infinite series of chiral or regular polytopes of rank 4 whose facets or vertex figures are toroidal maps. This is done by the use of the groups \(\text{PSL}_ 2 (\mathbb{Z}_ m)\), \(\text{PSL}_ 2 (\mathbb{Z}_ m [i])\) and \(\text{PSL}_ 2 (\mathbb{Z}_ m [\omega])\), where \(i^ 2 = - 1\) and \(\omega^ 2 + \omega + 1 = 0\). The types of these polytopes correspond to regular honeycombs in hyperbolic three-space. Each of the five types \(\{4,4,3\}\), \(\{4,4,4\}\), \(\{6,3,3\}\), \(\{3,6,3\}\), \(\{6,3,6\}\) is studied in detail. A final section concerns the lengths of Petrie polygons of regular or chiral polytopes. The paper constitutes a continuation and elaboration of previous work of the authors, published in the `Victor Klee Festschrift' [Discret. Math. Theor. Comput. Sci. 4, 493-516 (1991; Zbl 0741.51019)].
scientific article; zbMATH DE number 626340
scientific article