CHAPTER 3.7 - Geometric Crystallography

Abstract

This chapter focuses on geometric crystallography. Within a crystal, atomic building blocks, congruent to each other, are regularly arranged. Every discrete group of symmetry operations that acts transitively on a regular system of points in Ed is a d-dimensional space group. For each fixed d, there are many isomorphism classes of d-dimensional crystallographic groups. In contrast to point groups, space groups are affine equivalent only if they are isomorphic. In most cases, solidification of matter results in a periodic aggregation of an immense number of atoms or molecules. The Dirichlet domain is an important mathematical tool for investigating the metrical and topological properties of point sets. Domain of influence has applications in the determination of atom coordination numbers and in finding maximal holes in crystal structures. The metrical and topological properties of a set of points are revealed by the Dirichlet domain partition. The classification of Dirichlet domain partitions, with respect to symmetries, is done by affine mappings.