An Aperiodic Monotile Exists!(aperiodical.com)
aperiodical.com
An Aperiodic Monotile Exists!
https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/
52 comments
I have been waiting for this for 50 years! It is wonderful that Roger Penrose and my father, Alan Mackay, are still alive to see it! If only Escher could see it!
Neat! I'm impressed at its simplicity, after the complexity of the disconnected aperiodic Socolar-Taylor "tile" [1]. I am a bit disappointed it needs reflections though (the tiles are all the same shape but only fit together if you flip some of them over).
[1] https://en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile
[1] https://en.wikipedia.org/wiki/Socolar%E2%80%93Taylor_tile
Thanks for the link, this is very cool!
Weird (to me) that five out of six images show the 3D case, which is way harder to understand than a tile on a 2D plane, at least in my view.
It's also weird, for someone not at all familiar with the problem, that the concept of a "tile" includes non-connected shapes, i.e. there are outliers that are considered part of the tile but not connected to the main body. Apologies for absolutely no clue about terminology here.
Weird (to me) that five out of six images show the 3D case, which is way harder to understand than a tile on a 2D plane, at least in my view.
It's also weird, for someone not at all familiar with the problem, that the concept of a "tile" includes non-connected shapes, i.e. there are outliers that are considered part of the tile but not connected to the main body. Apologies for absolutely no clue about terminology here.
> It's also weird, for someone not at all familiar with the problem, that the concept of a "tile" includes non-connected shapes
The base concept here isn't the idea of a "tile". It's the idea of "tiling the plane", where "tile" is a verb. You tile the plane with a shape by repeating the shape over it and covering the whole space without leaving any holes or causing any overlaps.
Once you've defined the problem as "tiling", it's a natural step to call whatever shapes you happen to be using in your tiling scheme "tiles".
The base concept here isn't the idea of a "tile". It's the idea of "tiling the plane", where "tile" is a verb. You tile the plane with a shape by repeating the shape over it and covering the whole space without leaving any holes or causing any overlaps.
Once you've defined the problem as "tiling", it's a natural step to call whatever shapes you happen to be using in your tiling scheme "tiles".
I think the non-connected 'tile' was interesting simply because they hadn't yet found a connected one. When stuck solving a problem it's often helpful to look at what solutions would look like if some of the constraints were relaxed.
But in fact, the connected solution ended up much simpler than the disconnected one!
But in fact, the connected solution ended up much simpler than the disconnected one!
I'll be redoing my bathroom this year. What are the chances that some company will begin making these tiles in quality ceramic?
Find you a local pottery. Tiles are totally something you can still do in smallish quantity by hand. Mind that they won't be "calibrated" tiles though so it'll be a bit trickier to lay them neatly.
EDIT: Be sure to get mirror copies made too. It'd be a shame to go through all the effort and end up with the backside of tiles showing.
EDIT: Be sure to get mirror copies made too. It'd be a shame to go through all the effort and end up with the backside of tiles showing.
I have hand-made tiles in my kitchen in the house I just bought. They are definitely not "calibrated". :(
I just noticed that some of the tiles need to be placed up-side down. In a strict sense it consists of two tiles that eachother mirror.
I guess the strongest possible result would be a tile that can cover the plane aperiodically without reflections, but which can't cover the plane periodically even with reflections.
Or a proof that it's not possible.
I read this just after commenting that I'd like to use such tiles for my new bathroom. So indeed it would be two different tiles.
This image from the authors' webpage shows the hexagons formed by the 'kites', which makes it easier to understand how the tiling works: https://cs.uwaterloo.ca/~csk/hat/examples/hats.png
The colours aren't actually part of the solution, right? That's just to make it easier to visually comprehend?
Correct, the colors are not part of the tiling. They showi some additional structure which is used in the proofs of aperiodicity (and tiling). That said, the coloring is chosen in relation to the dark blue tiles, which are reflected relative to all other tiles.
I think they show the clusters? Light/dark blue = H, gray = F, white = T or P. Not sure, seems like you’d want one color per cluster rather than two colors for one and one color for two.
I don't have anything valuable to add but there's no comments here, so... I just wanted to say this is cool!
Previously on HN: https://news.ycombinator.com/item?id=35242458
This could make for an interesting board game mechanic hmmm
Great minds think alike :) https://news.ycombinator.com/item?id=35245511
The "Truchet-like markings" image is especially inspiring. I do want to try out to print out some tiles to play around
The "Truchet-like markings" image is especially inspiring. I do want to try out to print out some tiles to play around
Who else had "Anathem" by Stephenson come to mind?
Haha, came here looking for the Anathem comment!
Fraa Orolo can finally rest in peace* ;-)
* (Well ofcourse no spoilers)
Fraa Orolo can finally rest in peace* ;-)
* (Well ofcourse no spoilers)
I am amused that they call the Monotile an "einstein" ... because that is German for "one stone".
That's exactly why they named it like that.
Yes, it's good pun. That is what's amusing. No-one is taking it for a random coincidence.
"they call the Monotile an "einstein" because that is German for "one stone"" - does this sentence not say that this is "exactly why they named it that"?
"they call the Monotile an "einstein" because that is German for "one stone"" - does this sentence not say that this is "exactly why they named it that"?
It's ambiguous; you might have meant only that this was why you were amused, rather than why they named it so.
Clarity is hard in text; I have a few threads where someone carefully explains to me the joke or observation that I just made :D
Clarity is hard in text; I have a few threads where someone carefully explains to me the joke or observation that I just made :D
Clarity in writing is good in most cases, but necessarily not always so in humour: Strategic ambiguity implying both meanings is a frequent element of humour.
That was my intent.
I'm sorry if this fell flat for some.
That was my intent.
I'm sorry if this fell flat for some.
It says “what a coincidence”, and perhaps “how did they not notice?!” in a sort of mocking way.
In particular the “because” reads as applying to why this is amusing, not why they named it so.
Yes, it fell flat.
In particular the “because” reads as applying to why this is amusing, not why they named it so.
Yes, it fell flat.
> It says “what a coincidence”, and perhaps “how did they not notice?!”
I did not say any of that. Not even close. If you read that, that's you.
I did not say any of that. Not even close. If you read that, that's you.
You seemed genuinely interested in why you were not understood, so I was explaining my perspective. You can either get better at communicating or not, at your discretion.
I direct you to my earlier comment as to when clarity in communication is and is not the aim: https://news.ycombinator.com/item?id=35275136
And me
Someone please come up with a better name for this tile! (T_T)
Like an 8-kite, shirt-tile, or an origami tile...
Like an 8-kite, shirt-tile, or an origami tile...
What are the real life usages of this?
If nothing else, you could use the shape for floor tiles.
And if you use them in your house, future archeologists could definitely date your ruins to no earlier than 2022/2023. If you had a document and wanted to prevent it from being “backdated” earlier than 2022/2023, you could use the tiling pattern as a background. A geometric time stamp of sorts.
Either that, or it would upend the future understanding of our progress in geometry if you print it with old ink on old stock paper that’s dated substantially earlier than 2022/23.
Very interesting idea! We just have to be sure that these tiles don't already exists in some crystals or materials, in biological creatures, or somewhere else.
Sadly floor tiles have clear up and down sides, and this tiling requires them to be flipped now and then.
But if you could also get them in four colors, it'd still be a kind of ultimate mathematical floor.
But if you could also get them in four colors, it'd still be a kind of ultimate mathematical floor.
Procedural generation in games: Dungeon/Map generation or for combining or creating textures. And you could also used this maybe as an alternative to grid or hexagonal based games.
What makes this stand out is, that you can create larger structures of simple tiles, where repeating patterns and seams between tiles are less visible.
What makes this stand out is, that you can create larger structures of simple tiles, where repeating patterns and seams between tiles are less visible.
Imagine, gods forbid, getting lost in such a labirynth... 8(
Exactly this probably doesn't have any, but better math leads to better matter/material in general.
Trolling... I mean, throttling video cards ) Making untiling textures.
Also, this probably could have uses in watermarking and encryption (if you want to do so, I'm not patenting this, just give credits :D )
Also, this probably could have uses in watermarking and encryption (if you want to do so, I'm not patenting this, just give credits :D )
To optimize surface use in industrial print plants, orders are gathered, and they need to be combined in some way.
Not sure this particular tiling can help, but this popped immediately in my mind.
Not sure this particular tiling can help, but this popped immediately in my mind.
It could be a post-GPT watermark. If a document used this tiling pattern as a (faint) background, you could say definitively that the document was produced in the GPT era and thus might simply be automated babble.
Will it be possible to make crystal structures from this?
Camouflage fabric ?
Art