An Aperiodic Monotile (2023)

[zokier]

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/

[nhatcher]

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/

[Gare]

> 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

[gilleain]

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

[rini17]

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.

[zem]

it feels like it would be hard for those to tile at all, let alone aperiodially

[ganzuul]

Spiral out and keep going?

[yayamo]

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

[joelthelion]

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?

[shrx]

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

[bluepoint]

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

[bradrn]

(2023)

[mkesper]

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