Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

Abstract.

A new interactive software tool is described, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) originated as a script for the open source visual CLUCalc, which fully supports geometric algebra computation.

Selected generators (Hestenes and Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes.

We first give some insights into the Clifford geometric algebra description of space groups. We relate the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography (T. Hahn, Springer, 2005). Finally we briefly explain how to use the SGV beginning with space group selection. The interactive computer graphics can be used to fully understand how reflections combine to generate all 230 three-dimensional space groups.