Reciprocal Space and Crystal Planes in Geometric Algebra

This contribution discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space ${\mathbb{R}}^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representati on are related (and geometrically interpreted) in the projective geometric algebra ${\mathbb{R}}_{n+1}$ (see H. Grassmann, edited by F. Engel, Sie Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner, Leipzig, 1894.) and in the conformal algebra ${\mathbb{R}}_{n+1,1}$. The crystallographic notions of $d$-spacing, phase angle (in structure factors), extinction of Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context.