Coming to a bathroom floor near you soon
March 20, 2023 8:08 PM   Subscribe

An aperiodic monotile.

An aperiodic tiling (wiki) is a tiling made from the same basic elements or tiles that can cover an arbitrarily large surface without ever exactly repeating itself.
In 1962, the mathematician Robert Berger discovered a set of 20,426 tiles that did the trick.
Over time the number of tiles was reduced until Roger Penrose, in the early seventies, got it down to two.
While an einstein (ein stein=one tile) was almost achieved in 2010, that shape requires additional markings or modifications to tile aperiodically, which can't be encoded purely in its outline.
Today, in the paper linked at the top, David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss revealed a more natural solution.
Behold, The Hat! (Twitter link)
posted by thatwhichfalls (38 comments total) 63 users marked this as a favorite
 
SO COOL
posted by clew at 8:30 PM on March 20, 2023 [1 favorite]


Greg Kuperberg is going to need another bathroom.
posted by kaibutsu at 8:39 PM on March 20, 2023 [8 favorites]


The thing that blows me away is how simple the shape is. It's just four copies of a third of a hexagon glued together. I'm trying to get through the paper but honestly I just don't have the background or brain cells to make sense of it.
posted by phooky at 9:20 PM on March 20, 2023 [8 favorites]


To me it doesn’t look like it’s its own mirror image, but I don’t have much trust in my ability to see that even for such a relatively uncomplicated shape.

But that would mean they’ve discovered two 'ein stein' aperiodic tilings of the plane.
posted by jamjam at 11:22 PM on March 20, 2023 [2 favorites]


Fascinating, I'm looking fwd to burrowing in. Sort of related to an FPP I did last year on semi-irregular tiling.

The Socolar solution look like a kind of meta-tile (surface not pattern), I mean the shape is a true hexagon. I lack the math language to grasp this but it doesn't seem like a Penrose or Cairo tile.
posted by unearthed at 12:29 AM on March 21, 2023


Mastodon version of the Twitter thread, for those who don't want to engage with Twitter.

The tile is so simple-looking, but I guess the pudding was in the proof.
posted by sixohsix at 12:48 AM on March 21, 2023 [13 favorites]


The interactive gizmo is neat
posted by chavenet at 1:53 AM on March 21, 2023


I'm not grouting that.
posted by dowcrag at 3:00 AM on March 21, 2023 [18 favorites]


(goes down rabbit hole) Penrose Tilings, I wonder if ... wow yes he is the nephew of Surrealist Roland Penrose, husband of model and photographer Lee Miller ...
posted by GallonOfAlan at 3:39 AM on March 21, 2023 [2 favorites]


And they're from UWaterloo! Semi-hometown pride.
posted by saturday_morning at 4:31 AM on March 21, 2023 [3 favorites]


You can also play with it at Mathigon. Via Mathstodon
posted by vacapinta at 4:54 AM on March 21, 2023 [3 favorites]


If periodic tiling is the subject, I know Craig will be involved somehow. He's great.

Be sure to pick up his mobile game Good Fences if you like fucking with tiling in a playing-solitaire fashion.
posted by seanmpuckett at 6:17 AM on March 21, 2023 [5 favorites]


That's not a hat, it's a jersey (or hockey sweater)!
posted by wenestvedt at 6:41 AM on March 21, 2023 [2 favorites]


I would love to tile all my bathrooms with mathematically interesting tilings. I was a little disappointed when we ended up refreshing our bathrooms that the tiling was the one thing we didn't redo.

On the other hand, how hard would it be to find a contractor that would actually do it right? I'd probably have to learn how to tile myself and I am not good at that sort of thing.
posted by madcaptenor at 7:06 AM on March 21, 2023 [2 favorites]


Inkscape➡SVG file➡local library laser cutter!
Fun weekend ahead!
posted by Marky at 7:57 AM on March 21, 2023 [2 favorites]


Not just 1 monotile, but an infinite number of shapes along a continum. So if you don't quite like the hat there are other options for your bathroom.
posted by joeyh at 8:20 AM on March 21, 2023 [3 favorites]


I'm curious about whether or not it counts as a monotile since you have to use the tile and its mirror image. Were you to make actual tile for your bathroom you'd have to make two shapes because of the glazed and un-glazed sides. I wonder if any of the shapes along the continuum are symmetric with their mirror images, like Penrose kites and darts, or if that's even possible with aperiodic monotiles.
posted by indexy at 8:46 AM on March 21, 2023 [1 favorite]


That's not a hat, it's a jersey (or hockey sweater)!

It is a baby onesie, clearly.
posted by vacapinta at 9:01 AM on March 21, 2023 [1 favorite]


I'm curious about whether or not it counts as a monotile since you have to use the tile and its mirror image

Mathematically, reflections are allowed because the four isometries of the Euclidean plane are translation, rotation, reflection, and glide reflection - and symmetry is built on top of that definition. You can also investigate other sets of rules like disallowing reflection and many problems will have different or unknown answers. Based on the definition of the family of aperiodic shapes in the paper I don't think any of them would be symmetric with their mirror images (achiral).

Bringing these abstract tiling problems into a physical interpretation is always a bit weird because the aperiodic tile has to tile the infinite plane, which is somewhat larger than most bathroom floors.
posted by allegedly at 9:19 AM on March 21, 2023 [10 favorites]


Thanks, allegedly, it's been a while since I've studied this stuff and couldn't recall all of the parameters. But it makes more sense now and I'm remembering how things like the wallpaper groups are constructed.
posted by indexy at 9:31 AM on March 21, 2023


Alternative board game tiles apart from the traditional squares and hexagons?
posted by Harald74 at 11:16 AM on March 21, 2023 [1 favorite]


Alternative board game tiles apart from the traditional squares and hexagons?

Carcassonne is gonna get WILD!
posted by wenestvedt at 11:33 AM on March 21, 2023 [3 favorites]


I would love to tile all my bathrooms with mathematically interesting tilings. I was a little disappointed when we ended up refreshing our bathrooms that the tiling was the one thing we didn't redo.

On the other hand, how hard would it be to find a contractor that would actually do it right? I'd probably have to learn how to tile myself and I am not good at that sort of thing.


We are building a house right now and I have had this exact thought!
posted by TedW at 11:50 AM on March 21, 2023 [1 favorite]


Me trying to lay out copies of this shape for laser-cutting: "There has to be a way to fit more of them into the cuttable area!"
posted by Phssthpok at 12:05 PM on March 21, 2023 [2 favorites]


> Bringing these abstract tiling problems into a physical interpretation is always a bit weird because the aperiodic tile has to tile the infinite plane, which is somewhat larger than most bathroom floors.

Yeah but they could work for a ROUND bathroom right? Wait that's not infinite either. How about a spherical bathroom? OMG I need to talk to an architect!
posted by flamewise at 12:32 PM on March 21, 2023


Round bathroom floor, projective geometry, lifetime of glory for the tiler.
posted by clew at 12:38 PM on March 21, 2023 [5 favorites]


On the other hand, how hard would it be to find a contractor that would actually do it right? I'd probably have to learn how to tile myself and I am not good at that sort of thing.

We selected square tiles for two of our bathrooms that have pattern-looking designs on them but aren't meant to make any pattern. When we were in our house looking at how construction was going the tiler came up to us and asked us what pattern the tiles were supposed to make because he couldn't figure it out and I felt so bad when I had to tell him that there was no pattern and he could just install them randomly because I know he had spent time trying to lay them all out properly.
posted by any portmanteau in a storm at 3:06 PM on March 21, 2023


That suggests that I could probably arrange squares in an interesting pattern and get someone to do it in tile.
posted by madcaptenor at 3:19 PM on March 21, 2023 [1 favorite]


You definitely can as long as you're willing to pay them!

When I was tile shopping for my house I had the idea of a service that would take a user provided image, pixelate it, and then have a robot arm take that image and place the appropriate tiles onto 30cmx30cm backing sheets so that you could then tile the image at home without having to set each specific tile in place. I don't think it's a terrible idea but I could never picture it being more than an Etsy side-hustle.
posted by any portmanteau in a storm at 3:34 PM on March 21, 2023 [1 favorite]


you might be onto something, madcaptenor ...
posted by scruss at 6:27 PM on March 21, 2023


I read this news a couple days ago when Craig Kaplan posted it to Mastodon, and I couldn't rest until I got this terrible photoshop out of my system: https://mathstodon.xyz/@3j0hn/110065427944523246
posted by 3j0hn at 7:28 AM on March 22, 2023 [1 favorite]


This post at the APeriodical is a good overview and summary.
posted by vacapinta at 10:11 AM on March 22, 2023 [4 favorites]


I know there's a tiler on Craig Kaplan's site, but if you want to 3d print or laser cut the tiles to play with, it's a little fiddly.

Here's a not-too-small tile outline to play with: new-hat_tile.svg, made with OpenSCAD:
module hat(r) {
    // coordinates of all 13 vertices
    // output is shifted so no coordinates are negative
    translate([3 * r / 4, r * sqrt(3) / 2])polygon([
        [ 0,              0],
        [ 0,             -r * sqrt(3) / 2],
        [ r / 2,         -r * sqrt(3) / 2],
        [ r,              0],
        [ 3 * r / 4,      r * sqrt(3) / 4],
        [ 3 * r / 2,      r * sqrt(3) / 2],
        [ 3 * r / 2,      r * sqrt(3)],
        [ r,              r * sqrt(3)],
        [ 3 * r / 4,      3 * r * sqrt(3) / 4],
        [ 0,              r * sqrt(3)],
        [-3 * r / 4,      3 * r * sqrt(3) / 4],
        [-r / 2,          r * sqrt(3) / 2],
        [-3 * r / 4,      r * sqrt(3) / 4]
    ]);
}

hat(20);
posted by scruss at 2:01 PM on March 22, 2023 [4 favorites]


This is very exciting! I woled (wowed out loud).

Unfortunately, not as easy to cut in stained glass as the Penrose rhombs. But that's not going to stop everybody.
posted by inexorably_forward at 3:42 PM on March 22, 2023


SO COOL.
posted by rmd1023 at 3:52 PM on March 22, 2023


Unsurprisingly, people are already putting models up on thingiverse
posted by rmd1023 at 4:05 PM on March 22, 2023


and even where the cool kids hang out, on Printables: Search: hat monotile | Printables.com
posted by scruss at 8:49 AM on March 23, 2023 [1 favorite]


YouTube video by the authors in which they cover a brief history of tiling and talk about the result and why it is aperiodic.
posted by vacapinta at 1:32 AM on March 27, 2023 [4 favorites]


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