Abstract: The study of highly symmetric structures in Euclidean 3-space has a long and fascinating history tracing back to the early days of geometry. With the passage of time, various notions of polyhedral structures have attracted attention and brought to light new exciting figures intimately related to finite or infinite groups of isometries. A radically different, skeletal approach to polyhedra was pioneered by Grunbaum in the 1970’s building on Coxeter’s work. A polyhedron is viewed not as a solid but rather as a finite or infinite periodic geometric edge graph in space equipped with additional polyhedral super-structure imposed by the faces. Since the mid 1970’s there has been a lot of activity in this area. Much work has focused on classifying skeletal polyhedra and skeletal complexes by symmetry, with the degree of symmetry determined by distinguished transitivity properties of the geometric symmetry groups. These skeletal figures exhibit fascinating geometric, combinatorial, and algebraic properties and include many new finite and infinite structures. The talk gives an introduction to this area.
