An Aperiodic Monotile

[Guenther (Guenther Shepherd)]

Thanks for sharing your Specter GH file. I enjoyed getting to play around with making some of my own tile decorations based off your script.
Here’s just a few of the designs I came up with.

I’ll be dumping the rest of my designs on my personal website later.

Also, @bob.h.mackay I have a suggestion on how to position subcomponents more efficiently: Use the underlying hex tiling of specter for your positing.

Here’s a bit more of an explination as to what I’m thining. Referencing the Specter paper:

2.1 Main Result
[…] hexagons tile the plane, and from any such tiling we may construct a combinatorially equivalent tiling of the plane by Spectre

And those plane-tiling hexagons are equivelent to clusters of tiles like these:

In large tilings of Tile (1,1) the underly hex pattern of those little clusters is still visually apparent.

Maybe this could help find that different origin you needed? I apologize for only offering an idea and not actually implementing this, but that’s my suggestion.

[bob.h.mackay (Bob Mackay)]

Thanks for this suggestion post. I am afk at the moment, but will be on it when I get back.

Bob

[bob.h.mackay (Bob Mackay)]

It turns out that something like the Einstein aperiodic tiling may be happening in real life, at the molecular level. The inevitable Sabine Hossenfelder video gives a popularised view of this: Einstein-Tile Discovered in Nature. Maybe!

[maje90 (Riccardo Majewski)]

Aperiodic monotile is indeed interesting, but for real life scenarios (in real life scale), it still seems hard to use:

@DanielPiker

@maje90

Is there a smart way to check everything is going to be good, while putting down tiles one by one?
If one have to follow a printed/paper layout while putting tiles, we already have lost half of the usefulness of the aperiodic tile… or not?

That’s a good question.
Putting together a large patch by manually placing individual tiles without a hierarchical approach certainly becomes a very tricky puzzle.
For Penrose tiles there are the Ammann bars, as mentioned earlier in this thread, which can be used to decorate the tiles and determine the tiling.
For the monotile though (either the one with flips or this new one) there isn’t anything similar… yet!
The preprint only went up a few hours ago though. I remain hopeful that something like this will be found.