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/

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.

Good writeup of 'Combinatorial coordinates for the aperiodic Spectre tiling' from Simon Tatham here : https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...

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

Not really, since you can take a standard square tiling and apply a random shear transformation to it. With probability 1, you get a tiling of parallelograms with irrational angles.

Spiral out and keep going?

Interestingly this was found by a “hobbyist tiler”, David Smith, who is the first author. He was interviewed on how he found it in this YouTube video: https://youtu.be/4HHUGnHcDQw?si=VsHLqVUdw6ihERg2

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?

They discovered both variants, first the "Hat" and "Turtle" which require reflections, and then the "Spectre" which does not.

Does anyone of if there are any consequences of the existence of monotiles in algebra or number theory?

(2023)

There are new materials on the linked page like follow-ups and interactive applications.