New Views of Crystal Symmetry

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 (§ 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. Grassmann’s work in turn influenced W.K. Clifford (1878) in England: I propose to communicate in a brief form some applications of Grassmann's theory ... I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science. Conformal Clifford (geometric) algebra has in turn led at the beginning of the 20th century to a new fully geometric description of crystal symmetry in terms of so-called versors (Hestenes & Holt, 2007). Versors are simply (Clifford) geometric products of five-dimensional vectors conformally