Skeletal Polyhedral Geometry and Symmetry

Egon Schulte
Northeastern University

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Minisymposium: POLYTOPES AND GRAPHS

Content: Highly symmetric polyhedra-like structures in ordinary euclidean 3-space have a long history of study tracing back to the early days of geometry. With the passage of time, various notions of polyhedra have brought to light new exciting figures intimately related to finite or infinite groups of isometries. In the 1970's, a new ``skeletal" approach to polyhedra was pioneered by Gr\"unbaum building on Coxeter's work. We survey the present state of the classification of discrete skeletal structures by distinguished transitivity properties of their symmetry groups. These figures are finite, or infinite periodic, geometric edge graphs equipped with additional polyhedra-like structure determined by the collection of faces (simply closed planar or skew polygons, zig-zag polygons, or helical polygons). Particularly interesting classes of skeletal structures include the regular, chiral, or uniform skeletal polyhedra, as well as the regular polygonal complexes.

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