I don't know the physics well enough, but having skimmed the paper it looks like it has something to do with crystallographic point groups[1]. The point sets a

I don't know the physics well enough, but having skimmed the paper it looks like it has something to do with crystallographic point groups[1]. The point sets are invariant under some of these groups, and that symmetry can help you do some calculations. They also seem to make some use of dualities of the solids, which correspond to polyhedra with the same point group. It might have been more interesting to consider all of the point groups, but, again, I don't understand the physics to know why they did what they did. [1] https://en.wikipedia.org/wiki/Crystallographic_point_group https://en.wikipedia.org/wiki/Crystallographic_point_group