Representation of Crystallographic Subperiodic Groups in Clifford's Geometric Algebra

TL;DR

This paper extends the representation of 3D crystallographic space groups to all 162 subperiodic groups using Clifford's geometric algebra, introducing a new compact symbolic notation and explicit generators.

Contribution

It introduces a novel geometric algebra group representation symbol for subperiodic groups, enabling easier reading of generators and explicit representation.

Findings

Unified geometric algebra representation for all subperiodic groups

Explicit generators for each group in the new notation

Facilitates analysis of frieze, rod, and layer groups

Abstract

This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford monomials, Lipschitz elements). Keywords: Subperiodic groups, Clifford's geometric algebra, versor representation, frieze groups, rod groups, layer groups .