×

Chiral polyhedra in ordinary space. I. (English) Zbl 1059.52020

This article classifies the polyhedra with finite facets and vertex figures which are geometrically chiral in 3-dimensional space. An abstract polytope is regular if its automorphism group is transitive on the set of its flags. It is chiral if the automorphisms split the flags into two orbits in such a way that adjacent flags are in different orbits. A realisation of an abstract polytope is geometrically regular or geometrically chiral if similar statements may be made about its symmetry group. If the realisation of the polytope is geometrically regular, the polytope must also be regular. Likewise, if the realisation of the polytope is geometrically chiral, the polytope must be chiral or regular.
This article is the first part of a two part series. The goal of the series is to classify the discrete geometrically chiral polyhedra in 3-dimensional euclidean space.
This article shows first that such a polyhedron must be infinite. Then, assuming the facets and vertex figures are finite, it shows that they must be either squares or hexagons - in fact, the Schläfli type of the polyhedron must be {6,6}, {6,4} or {4,6}. The cases {6,6} and {4,6} are then considered in detail (those of type {6,4} will be the duals of the latter). The approach in both cases is similar.
A family of examples of chiral polytopes of the appropriate Schläfli type is constructed. This family is shown to be exhaustive of chiral polytopes of that type. Then, detailed information about the geometry of the family members is derived.
It remains an open question whether the geometrically chiral polytopes uncovered are abstractly chiral or regular.

MSC:

52B15 Symmetry properties of polytopes
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
PDFBibTeX XMLCite
Full Text: DOI