New Views of Crystal Symmetry

TL;DR

This paper explores the historical development of crystal symmetry understanding, highlighting Grassmann's vectorial approach and its influence on Clifford's geometric algebra, leading to modern computational methods.

Contribution

It demonstrates how Grassmann's early work on crystal forces influenced Clifford's geometric algebra, enabling a new computational approach to crystal symmetry.

Findings

Grassmann's vectorial system models crystal interior forces

Clifford's geometric algebra extends crystal symmetry description

Modern computer algebra graphics benefit from this geometric framework

Abstract

Already Hermann Grassmann's father Justus (1829, 1830) published two works on the geometrical description of crystals, influenced by the earlier works of C.S. Weiss (1780-1856) on three main crystal forces governing crystal formation. In his 1840 essay on the derivation of crystal shapes from the general law of crystal formation Hermann established the notion of a three-dimensional vectorial system of forces with rational coefficients, that represent the interior crystal structure, regulate its formation, its shape and physical behavior. In the Ausdehnungslehre 1844 (Paragraph 171) he finally writes: I shall conclude this presentation by one of the most beautiful applications which can be made of the science treated, i.e. the application to crystal figures (Scholz, 1996). The geometry of crystals thus certainly influenced the Ausdehnungslehre. In this paper we see how Grassmann's work influenced Clifford's creation of geometric algebras, which in turn leads to a new geometric description of crystal symmetry suitable for modern computer algebra graphics.