E.S.M. Hitzer, C. Perwass

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

Adv. Appl. Clifford Alg.,
Vol. 20(3-4), pp. 631-658, (2010),
DOI 10.1007/s00006-010-0214-z.

**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) is 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.

**Mathematics SubjectClassification (2000).**
Primary 20H15;
Secondary 15A66, 74N05, 76M27, 20F55 .

**Keywords.**
Clifford geometric algebra, interactive software, space groups, crystallography,
visualization.

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