Skeletal Polyhedra, Geometric Graphs, and Crystallographic Groups

The study of highly symmetric geometric structures in ordinary 3-dimensional Euclidean space has a rich history tracing back to the early days of geometry. Over the course of time, new notions of discrete polyhedral structures have intrigued scientists and brought to light new exciting figures intimately related to finite or infinite groups of isometries.

A new graph-theoretical, or skeletal, approach to polyhedral structures was pioneered by Branko Grunbaum in the 1970's and built on Coxeter's extensive work on polyhedra, polytopes, and symmetry. Skeletal polyhedra or polyhedral complexes in ordinary space are viewed not as 3-dimensional solids but rather as finite or infinite periodic geometric graph, linearly embedded into space and equipped with an overarching polyhedral super-structure imposed by the faces.

This new definition was deliberately designed to rid the theory of polyhedra of the psychological block that membranes must be spanning the faces to give a surface. In the new framework, the familiar 3-dimensional cube is viewed as a graph with eight vertices, twelve edges, and six 4-cycles in the graph distinguished as faces, namely the boundary cycles of the usual square of the cube. The level of symmetry of skeletal polyhedra or complexes is determined by distinguished transitivity properties of their geometric symmetry groups, which are either finite groups of isometry and infinite crystallographic groups of isometries.

These skeletal figures in ordinary space exhibit fascinating geometric, combinatorial, and algebraic properties and include both new and well-known structures. We discuss approaches to the classification for some of the most prominent classes of skeletal figures including the regular, chiral, or uniform polyhedra. The talk gives an introduction to this area.