The reflection group of the regular tetrahedron corresponds to the group of permutations of {1,2,3,4}. Shown are three special reflections that interchange 1&2, 2&3, and 3&4.

In how many ways can we place q Queens on an nxn chessboard so that no two attack each other?  We use Ehrhart theory to answer this question by counting lattice points inside a dilating polygon.  

For every ion in a molecule, one can calculate the force it feels from all the other ions in the molecule.  This value is the ion's Madelung Constant.  Shown is a nanotube with 8 layers of 30 ions.