3 ms·

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. h

• nhatcher 2y ago
  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/ https://www.nhatcher.com/post/on-hats-and-sats/
• Gare 2y ago
  > 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
  • foobarbecue 2y ago
    I, for one, would really like a spectre soccer ball. Seriously, though, I think the implications for mineralogy are interesting.
    • robinhouston 2y ago
      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 https://youtube.com/shorts/_Rruxxrz9nY
      • foobarbecue 2y ago
        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?
        • gilleain 2y ago
          Well you cannot tile a sphere with just hexagons, you need a minimum of 5 pentagons. Oh, from https://en.wikipedia.org/wiki/Fullerene 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.
        • zokier 2y ago
          I would be really surprised if hat (or spectre) could tile sphere. Afaik most tilings of planes do not work on spheres.