Colorings of patterns fixed by an arbitrary finite-index subgroup of the symmetry group.

In this research project, a framework is established that characterizes the colorings of a pattern fixed by an arbitrarily chosen finite-index subgroup of the pattern's symmetry group. This framework is applied to the sphalerite structure to enumerate all colorings fixed by two different subgroups of the symmetry group. These colorings are used to perform ordered substitution on the atoms in the sphalerite structure and demonstrate some of the well known derived structures (e.g. chalcopyrite, stannite). Some results involving equivalences of colorings fixed by the chosen subgroup are also developed.