New views of crystal symmetry guided by profound admiration of the extraordinary works of Grassmann and Clifford

TL;DR

This paper explores the historical influence of crystallography on Grassmann and Clifford, leading to advanced geometric algebra frameworks that enhance visualization of crystallographic space groups in computer graphics.

Contribution

It introduces a novel visualization method for 3D crystallographic space groups using conformal geometric algebras inspired by historical mathematical developments.

Findings

New visualization of space groups created

Enhanced understanding of geometric algebra applications

Historical influence on modern mathematical frameworks

Abstract

This paper shows how beginning with Justus Grassmann's work, Hermann Grassmann was influenced in his mathematical thinking by crystallography. H. Grassmann's Ausdehnungslehre in turn had a decisive influence on W.K. Clifford in the genesis of geometric algebras. Geometric algebras have been expanded to conformal geometric algebras, which provide an ideal framework for modern computer graphics. Within this framework a new visualization of three-dimensional crystallographic space groups has been created. The complex beauty of this new visualization is shown by a range of images of a diamond cell. Mathematical details are given in an appendix.