  This paper 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 representation are related (and geometrically interpreted) in the
projective geometric algebra $Cl(\R^{n+1})$ (see Grassmann H., edited by Engel
F., Die Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner:
Leipzig, 1894.) and in the conformal algebra $Cl(\R^{n+1,1})$. The
crystallographic notions of $d$-spacing, phase angle, structure factors,
conditions for Bragg reflections, and the interfacial angles of crystal planes
are obtained in the same context.
  Keywords: Clifford geometric algebra, crystallography, reciprocal space,
$d$-spacing, phase angle, structure factors, Bragg reflections, interfacial
angles
