An Aperiodic Monotile (2023) (opens in new tab)

(cs.uwaterloo.ca)

59 pointsphaedryx1y ago19 comments

19 comments

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/

nhatcher1y 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/

Gare1y 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

foobarbecue1y ago

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

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

1 more reply

gilleain1y ago

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

rini171y ago

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.

zem1y ago

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

WCSTombs1y ago

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.

ganzuul1y ago

Spiral out and keep going?

yayamo1y ago

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

joelthelion1y ago

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?

shrx1y ago

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

bluepoint1y ago

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

bradrn1y ago

(2023)

mkesper1y ago

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

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