After publication of Spectres, I don't know if there much interest anymore on Hats. Spectres are like Hats, but eliminate the need of reflections for tiling.
I did a write up with some app you can play with a while ago:
And people say that mathematical research has no practical applications
Seriously, though, I think the implications for mineralogy are interesting.
Oh, from https://en.wikipedia.org/wiki/Fullerene:
"A closed fullerene with sphere-like shell must have at least some cycles that are pentagons or heptagons. More precisely, if all the faces have 5 or 6 sides, it follows from Euler's polyhedron formula, V−E+F=2 (where V, E, F are the numbers of vertices, edges, and faces), that V must be even, and that there must be exactly 12 pentagons and V/2−10 hexagons. "
So I'm not sure.