  This contribution discusses the geometry of $k$D crystal cells given by
$(k+1)$ points in a projective space $\R^{n+1}$. 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 $\R_{n+1}$ (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 $\R_{n+1,1}$. 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.
