Characterization of patterns formed by shadows of spheres.

Motivated by colloidal lithography, we study the problem of characterizing periodic planar patterns formed by shadows of spheres. The set of patterns accessible to shadow lithography spanned by lattice types, tilt, and rotation angles is rich, but topological considerations of shadow overlap along simplex edges and faces lead us to just 4+1 distinct categories. These planar patterns are in one-to-one correspondence with a 4-valued index linked to Cayley-Menger determinants. The characterization is confirmed by a phase diagram which predicts surface patterns for any experimental geometry.