I think they are very interesting as a first step in the construction.

I did a write up with some app you can play with a while ago:

https://www.nhatcher.com/post/on-hats-and-sats/

> It also complicates the practical application of the hat in some decorative contexts, where extra work would be needed to manufacture both a shape and its reflection

And people say that mathematical research has no practical applications

I, for one, would really like a spectre soccer ball.

Seriously, though, I think the implications for mineralogy are interesting.

I don't know whether anyone's designed a spectre soccer ball, but someone has designed a soccer ball based on the hat tile.

https://youtube.com/shorts/_Rruxxrz9nY

Interesting. I wonder why the five pentagons were needed. Is that because hat can't tile a sphere? Or some other requirement when assembling the ball?

Well you cannot tile a sphere with just hexagons, you need a minimum of 5 pentagons.

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.

I would be really surprised if hat (or spectre) could tile sphere. Afaik most tilings of planes do not work on spheres.