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Chiral polytopes. (English) Zbl 0741.51019

Applied geometry and discrete mathematics, Festschr. 65th Birthday Victor Klee, DIMACS, Ser. Discret. Math. Theor. Comput. Sci. 4, 493-516 (1991).
[For the entire collection see Zbl 0726.00015.]
The authors present a careful study of chiral polytopes. These are certain abstract or incidence polytopes, as introduced by L. Danzer and the first author [Geom. Dedicata 13, 295-308 (1982; Zbl 0505.51019)]. A polytope in this sense is regular if its automorphism group is transitive on the flags. While abstract regular polytopes have been studied intensively, much less was known so far about chiral polytopes, the class of abstract polytopes with maximal symmetry by rotation. More precisely, a polytope \(\mathcal P\) of rank \(n\geq 3\), with a fixed flag \(\Phi=\{F_{-1},F_ 0,\dots,F_ n\}\), is called chiral if \(\mathcal P\) is not regular, but there are automorphisms \(\sigma_ 1,\dots,\sigma_{n- 1}\) of \(\mathcal P\) such that \(\sigma_ i\) fixes all faces in \(\Phi\setminus\{F_{i-1},F_ i\}\) and cyclically permutes consecutive \(i\)-faces of \(\mathcal P\) in the (polygonal) rank 2 section \(F_{i+1}/F_{i-2}\) of \(\mathcal P\). This paper discusses the basic theory of chiral polytopes.

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
52B15 Symmetry properties of polytopes
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