A Geometric Shape That Does Not Repeat Itself When Tiled - Slashdot
source link: https://science.slashdot.org/story/23/03/24/2232256/a-geometric-shape-that-does-not-repeat-itself-when-tiled
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A Geometric Shape That Does Not Repeat Itself When Tiled
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A Geometric Shape That Does Not Repeat Itself When Tiled (phys.org) 45
Posted by BeauHD
on Saturday March 25, 2023 @10:34AM from the aperiodic-monotile dept.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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Repeatedly. It just doesn't create repeating patters. Stupid title.
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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.
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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].
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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.-
The definition of an aperiodic tiling is that there is no translational symmetry, i.e. you can't move the tiling so that it would cover itself perfectly. For example, some Penrose tilings have rotational symmetry but they are still aperiodic.
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Think Penrose tiles. But done with a single shape.
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The decimal version of the binary result is https://oeis.org/A162145 [oeis.org] (after more thought). Perhaps more suitable for the discussion of the passing of Moore?
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Exactly.
But only if we consider a shape and its reflection as a single shape.
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And without rotational symmetry.
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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.
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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.
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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.
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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.
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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...
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Following this logic, you'll inevitably stub a toe. It might take miles, but you'll get there.
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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".
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At least it'd not be a repetitive job! =]
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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.
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It's no harder than any other grout job if you're doing it right.
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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.:)
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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.
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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.
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Sorry, I thought they were talking about some other imaginary shape.
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a lot of work when laying them.
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Sounds like an opportunity!
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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.
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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:)
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A Yorkshire university? The University of York? York University (which is not in Yorkshire)?
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Answering my own question, it's someone from Yorkshire not at any university.
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It could be intended to refer to https://yorkshireuniversities.... [yorkshireu...ties.ac.uk] (registered address in Leeds).
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That's just an umbrella organisation. I think TFA is confused by someone who is referred to in the paper as being from Yorkshire and with a Gmail address and there being a 'Yorkshire University'.
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If it can be leveraged into texture maps for CGI, that's kind of a holy grail solution to a significant problem.
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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]
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Be careful. The above link contains a browser attack
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Some kind of history overflow? I did not analyze, just closed the tab ASAP.
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Pretty sure that's BS, it's just a link to imgur.
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Will the theory be broken when/. re-posts this story next week
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Hmm, that looks like an invasive program that could take out the Borg once and for all...
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