11–20 of 21 posts
Something that is unclear to me: are hat reflections allowed? I think they are, but it would be good to have confirmation. In short, if you allow reflections, are the tilings still guaranteed to be aperiodic?
Next frontier: aperiodic tilings with irrational angles (meant, tiles having angles of x*2pi were x is irrational). Or are these proven to be impossible? Because both the hats and spectres are basically subset of triangular grid. Penrose tilings are subset of regular grid, too. Can we get rid of these underlaying regular grids.
it feels like it would be hard for those to tile at all, let alone aperiodially
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. https://cs.uwaterloo.ca/~csk/spectre/
> 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
Seriously, though, I think the implications for mineralogy are interesting.
Next frontier: aperiodic tilings with irrational angles (meant, tiles having angles of x*2pi were x is irrational). Or are these proven to be impossible? Because both the hats and spectres are basically subset of triangular grid. Penrose tilings are subset of regular grid, too. Can we get rid of these underlaying regular grids.
Earlier quoted context omitted.
> 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.
Earlier quoted context omitted.
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
Earlier quoted context omitted.
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?
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.
Earlier quoted context omitted.
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?