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Math

A Geometric Shape That Does Not Repeat Itself When Tiled (phys.org) 72

Posted by BeauHD from the aperiodic-monotile dept.
IHTFISP shares a report from Phys.Org: A quartet of mathematicians from Yorkshire University, the University of Cambridge, the University of Waterloo and the University of Arkansas has discovered a 2D geometric shape that does not repeat itself when tiled. David Smith, Joseph Samuel Myers, Craig Kaplan and Chaim Goodman-Strauss have written a paper describing how they discovered the unique shape and possible uses for it. Their full paper is available on the arXiv preprint server. [...]

The shape has 13 sides and the team refers to it simply as "the hat." They found it by first paring down possibilities using a computer and then by studying the resulting smaller sets by hand. Once they had what they believed was a good possibility, they tested it using a combinatorial software program -- and followed that up by proving the shape was aperiodic using a geometric incommensurability argument. The researchers close by suggesting that the most likely application of the hat is in the arts.
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A Geometric Shape That Does Not Repeat Itself When Tiled More

A Geometric Shape That Does Not Repeat Itself When Tiled

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  • Of course it repeates itself. (Score:3, Insightful)

    by fredrated ( 639554 ) on Saturday March 25, 2023 @10:44AM (#63398453) Journal

    Repeatedly. It just doesn't create repeating patters. Stupid title.

    • It's also not really new... isn't this just a reapplication of lines across the hexagonal vertices? They just seem to have found the set for single shape and lowest polygon count.

      Or am I missing something here? There must be other shapes with more polygons that meet this criteria.

      • Re: Of course it repeates itself. (Score:4, Informative)

        by sjames ( 1099 ) on Saturday March 25, 2023 @04:27PM (#63398991) Homepage Journal

        You are missing something. Go to TFA (first link). Look at the diagram. The grey area is the shape of a single tile. When laid out, the pattern is non-repeating. There is not a way to lay that shape out such that it repeats.

        The underlying hexagons and triangles do repeat. You can see that in the same diagram.

        This is the first such shape identified. Have a look here [wikipedia.org].

      • "I have no idea what's going on, but I'm pretty sure these smart people are wrong, and there must be something that proves me right. Or am I missing something here?"

    • Maybe I should look at a larger image, but I saw three pairs with identical positions in the article's image.
      I'd call that repeating.

    • Re: (Score:1)

      by jonadab ( 583620 )
      Yeah, but I thought the existence of a repeating pattern was necessarily implied by the definition of the word "tiled".

  • Quicker summary (Score:5, Insightful)

    by isj ( 453011 ) on Saturday March 25, 2023 @11:00AM (#63398467) Homepage

    Think Penrose tiles. But done with a single shape.

    • Re:Quicker summary (Score:4, Insightful)

      by leonbloy ( 812294 ) on Saturday March 25, 2023 @02:31PM (#63398813)

      Exactly.

      But only if we consider a shape and its reflection as a single shape.

    • And without rotational symmetry.

    • Re: (Score:2)

      by pr0nbot ( 313417 )

      I wonder how they prove it will work infinitely. Presumably there's no combination of more than one of these tiles that repeats periodically either.

  • DIY nightmare (Score:5, Funny)

    by Powercntrl ( 458442 ) on Saturday March 25, 2023 @11:03AM (#63398473) Homepage

    As someone who has laid flooring tile a few times, let me be the first to say: I'd hate to have to grout that.

    • Re:DIY nightmare (Score:5, Insightful)

      by Excelcia ( 906188 ) <slashdot@excelcia.ca> on Saturday March 25, 2023 @11:26AM (#63398533) Homepage Journal

      I'd hate to have to grout that.

      That's not the only problem. You would need special software to lay it, because it not only has the ability to have no repeating section, it also has the ability to make it so you can't add onto it any more. You could easily be laying tiles and have them fit nicely, only to find that you've created a design that doesn't actually work. Imagine getting 2/3rds of a room done, and discovering because of the way you permuted it ten tiles in you can't go any further. A shape like this that admits infinite many tilings that work, also admits infinite that don't.

      • These are both good points but they assure this will find a place as a luxury item.

        After I remove the icepack, I use a deep-pore cleanser lotion.
        In the shower, I use a water-activated gel cleanser.
        The tiles in my bathroom were made of volcanic glass collected from Sakurajima and cut by native craftsmen. Each one completely unique. Even the layout is aperiodic, that is that the patten never actually repeats.
        Every square inch of my bathroom is unique and special.

        The perfect bathroom.

      • Re: (Score:3)

        by Gherald ( 682277 )

        With Penrose tiles there's a simple rule about which edges you should line up that, if you follow it, allows you to tile an infinite plane without running into that problem of not being able to add on. So there's probably something analogous rule for this one tile version.

      • Re: (Score:2)

        by ebh ( 116526 )

        There are some examples in the full paper that would get you a big enough pattern for a very large room, given tiles that are at least the size of the palm of your hand. And the authors probably do have the code that would lay out as many tiles as you'd want.

        I'm actually in the last stages of a bathroom remodel. If I'd known about this a couple months ago...

      • Why does the person who has never laid tile and thinks tile laying tile involves software get "Insightful"? Oh yeah, because this isn't "News for Handymen".
      • The word you're looking for is "planning". It's complex, but the gist is - you make a (mathematical, computational, whatever) model of your planned deployment, check out your possible forms of tiling, then reject those that don't give you an appropriate result in your field of deployment. If the tiling goes wonky outside your work area, so what?

        A shape like this that admits infinite many tilings that work, also admits infinite that don't.

        Surely, a smaller infinity that don't work?

        I still struggle to see th

      • That’s when the tile saw comes out.

      • Re: (Score:2)

        by Briareos ( 21163 )

        Imagine getting 2/3rds of a room done, and discovering because of the way you permuted it ten tiles in you can't go any further.

        Dorfromantik manages to do the same with it's hexagonal tiles alone...

    • At least it'd not be a repetitive job! =]

    • Re: (Score:3)

      by dargaud ( 518470 )
      I always dreamed of having Penrose tiles on my wood floor. When I bought my 2nd home and had to do the flooring for the 2nd time, I looked into it. I found a guy in Germany who could do Penrose wood floor, it did look great but the price was too much for my budget. Next time.
      • Somewhere I've got the plans and dimensions for a two-parallelogram Penrose tiling on a "living room" scale (room less than 10m in either dimension) Cut the flooring into strips of two widths, then re-set your cutting table (or jig) to the correct angle to the "strips" and get the second cut in on the two different sets. While you're doing that, mark the tile edges with the "matching rules" (several variants possible) to guarantee aperiodicity, then start laying.

        The next time I need to tile a floor (or for

        • Re: (Score:2)

          by dargaud ( 518470 )
          I could probably do the cutting properly, and the positioning has software for that. But how do you do the interlocking part ? Or you don't and glue the tiles in place ? I don't understand your part about cutting strips.
          • The Penrose tiling I was looking at (one of several possibilities) was composed of two types of parallelograms (which are also rhombuses), "thin" and "fat". "Matching rules" can be indicated in various ways. If you're Mauritz Escher, you can turn your parallelograms into birds or something artistic. If you're writing the Wiki page [wikipedia.org], you can put circular arcs across the parallelogram corners, then link them between several parallelograms to make circles (sometimes across 3 parallograms meeting, sometimes acro

            • Re: (Score:2)

              by dargaud ( 518470 )
              OK, I get it now. The original wood strips would have to be fairly large in order to get decently-sized rhombuses. And you'd get at most 50% or working interlocks, the other sides would be just plain vertical cuts. Still, better than nothing and probably OK on a good flat underlayer. I've laid wood flooring on 6 or 7 rooms at various times, but I swear next time I won't let my wife order me around to the simplest and cheapest solution and I'll try it !!!
              • Laminate flooring comes in (typically) 1sq.m packs here, with about a 2:1 aspect ratio per sheet, and (I think) 8 sheets per pack. I should be able to work out the dimensions per piece from that, but I've not got close enough to doing the job yet. It'd be quicker to get a pack (5 minute walk from the front door) and measure it than calculate it.

                The smaller the rhombuses (rhombs, rhombii? whatever) you cut, the lower the wastage. So I think, maybe 2 strips of "fats" and 2 of "thins" from each sheet. The tri

                • Re: (Score:2)

                  by dargaud ( 518470 )
                  If you try it, send me some pics. Good luck.
                  • Right ; I've saved the discussion as a PDF (locally just now) down to this point. Shoved that onto a storage server (I really need to find a better place), uploaded a few images and description to my blog [blogspot.com]. But now I've got to go and do my French practice.

                    • Re: (Score:2)

                      by dargaud ( 518470 )

                      But now I've got to go and do my French practice.

                      Fallait le dire avant, on aurait pu faire d'une pierre deux coups... :-)

                    • Oh fuck slashdot and it's stupid cloudflare authorisation bullshit. Just because it takes a couple of minutes to complete the cloudflare crap, slashdot has thrown away my comment.

    • Re: (Score:2)

      by sjames ( 1099 )

      It's no harder than any other grout job if you're doing it right.

    • As someone who has laid tile a few times in old homes where nothing is square, being able to say "it's supposed to look like that" might be a plus. :)

      • As someone who not only lives in one of those old homes where nothing is square, but has also done copious renovations to said home, this gave me a good laugh. Thank you.

    • As a bathroom tile pattern, this would be easiest to simulate by just using kite-shaped tiles of multiple colors and arranging eight tiles of the same color to form each "hat" shape. The underlying kite shape is something you can already buy.

    • As someone who has laid flooring tile a few times, let me be the first to say: I'd hate to have to grout that.

      As someone else who has laid flooring tile a few times, I'm curious what problem you see? Applying grout is mostly a matter of pushing the grout down into gaps between the tiles, followed by a bit of wiping and cleanup. The orientation of the gaps is inconsequential.

  • Sorry, I thought they were talking about some other imaginary shape.

  • a lot of work when laying them.

  • Heat Shielding? (Score:5, Interesting)

    by bill_mcgonigle ( 4333 ) * on Saturday March 25, 2023 @12:12PM (#63398597) Homepage Journal

    Thermal tiles on a reentry vehicle with no stress line patterns?

    Can this be tessellated into a geodesic dome? I might have to relearn that math. A single-shape construction system would be nice if we're going to colonize planets and stuff.

    • No, this only works in a 2D plane. It couldn't be wrapped around a cylinder (because that requires the pattern to repeat) let alone a sphere. Non-repeatability is not an asset in construction. If you want to use a single shape, just use a triangle or a square :)

  • Yorkshire University (Score:3)

    by q_e_t ( 5104099 ) on Saturday March 25, 2023 @12:36PM (#63398623)
    A Yorkshire university? The University of York? York University (which is not in Yorkshire)?

  • If it can be leveraged into texture maps for CGI, that's kind of a holy grail solution to a significant problem.

  • Still very similar (Score:3)

    by Dwedit ( 232252 ) on Saturday March 25, 2023 @12:56PM (#63398655) Homepage

    Might not be a true 100% repeating pattern, but it looks extremely similar, and maybe close enough to count as such.

    Here is one placed on top of another at an offset position at 50% transparency:

    https://i.imgur.com/Z1jLq7W.pn... [imgur.com]

  • repeat (Score:3)

    by bugs2squash ( 1132591 ) on Saturday March 25, 2023 @01:27PM (#63398719)
    Will the theory be broken when /. re-posts this story next week

  • Hmm, that looks like an invasive program that could take out the Borg once and for all...

  • It would be really cool to have a ceramic tile design that was like jigsaw puzzle pieces. Yes, there are a few out there that are solid colors for each piece. But I'm thinking a jigsaw puzzle floor mural. It would be awful to actually lay such tile, but the effect would be awesome!

  • Could it be digitized to make a random number generator or be used with cryptography in some way?
  • When I think of "tiling" I consider it cheating to use a mirrored version of the shape. Why is that okay here?

  • I'd like to see a simple 4-coloring rule for it. Then one could not only tile the plane aperiodically with one shape, but do it with four colors such that each tile is surrounded by others that don't match it.

    A 4-coloring is guaranteed possible. But not guaranteed to be easy to compute.

  • Now I know how I am going to tile my kitchen.

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