An Aperiodic Monotile (2023)

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/

Re: An Aperiodic Monotile (2023)

#5

Post by nhatcher » Mon, Dec 02, 2024, 9:36 AM UTC

zokier wrote: post #4

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/

Re: An Aperiodic Monotile (2023)

#6

Post by rini17 » Mon, Dec 02, 2024, 9:51 AM UTC

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.

Re: An Aperiodic Monotile (2023)

#7

Post by zem » Mon, Dec 02, 2024, 10:03 AM UTC

rini17 wrote: post #6

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

Re: An Aperiodic Monotile (2023)

#8

Post by [deleted] » Mon, Dec 02, 2024, 10:18 AM UTC

[deleted]

Re: An Aperiodic Monotile (2023)

#9

Post by Gare » Mon, Dec 02, 2024, 10:24 AM UTC

zokier wrote: post #4

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

Re: An Aperiodic Monotile (2023)

#10

Post by joelthelion » Mon, Dec 02, 2024, 10:59 AM UTC

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?