Egon Schulte's talk at MACSMIN 2021 : Mathematics and Computer Science for Materials Innovation
Channel: MACSMIN conference 2021
Date: 2021-09-17

Title. Skeletal Polyhedra, Complexes, and Symmetry.
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 have brought to light new exciting figures intimately related to finite or infinite groups of isometries. A radically new, 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. The lecture surveys the present state of the ongoing program to classify discrete polyhedral structures in ordinary space by symmetry, where the degree of symmetry is defined via distinguished transitivity properties of the geometric symmetry groups. These skeletal figures exhibit fascinating geometric, combinatorial, and algebraic properties and include many new finite polyhedra as well as many new periodic structures with crystallographic symmetry groups.