13 ms·
Mathematicians discover shape that can tile a wall and never repeat
- iamben 4y agohttps://archive.is/iqBHP https://archive.is/iqBHP
- mnw21cam 4y agoCan someone ELI5 why this is different from Penrose tiling?
- timmg 4y agoAs I understand it: they found a single shape. Where existing Penrose tilings were composed of two (or more) shapes.
- phkahler 4y agoBut in one sense this is actually 2 shapes that are mirror images. It's still really cool, but I don't think it is ultimately what we've been looking for. As proof that it's not, we all know that a paper presenting one that doesn't need its mirror image to tile the plane aperiodically would still be a big dea.
- pohl 4y ago...they found a single shape Kind of, though, right? One could also look at it as they've found two shapes that happen to be reflections of each other.
- Jarmsy 4y agoRigid transformations of a single shape https://en.wikipedia.org/wiki/Rigid_transformation https://en.wikipedia.org/wiki/Rigid_transformation
- dekhn 4y agoIt's not really a single shape since the tiling contains the shape's reflection which is normally considered a different shape since its handedness changes.
- sojuz151 4y agoIt is just a single tile type, not two
- phkahler 4y agoThe two Penrose tiles are also an affine transformation of each other.
- Jarmsy 4y agoWhereas in this new tiling there's a single shape and its rigid transformations.
- OscarCunningham 4y agoI don't think they can be if you want to enforce the matching rules using the tile's shape.
- hgsgm 4y ago> matching rules using the tile's shape. That excludes affine transformation.
- lsaferite 4y agoThat's covered in the article.
- ggrelet 4y agoIt has a paywall.
- jlund-molfese 4y agoBut almost every HN post to pay-walled content includes an archive.is link to bypass the paywall.
- lsaferite 4y agoThey didn't indicate they couldn't read the article due to a paywall. They just asked for an ELI5 on something that's covered in the 4th paragraph of the article. My reaction would be different if you said you couldn't read the article and asked the same question.
- schiffern 4y agohttps://archive.is/RW7Wy https://archive.is/RW7Wy
- deleted 4y ago[deleted]
- LorenDB 4y agoIn the second image, the tiles look like West Virginia.
- mc32 4y agoRoughly it looks like a T-shirt. One with a jagged bottom. Jagged T-shirt tile.
- tromp 4y agoYes, this article goes more into the attire aspects: https://aperiodical.com/2023/03/an-aperiodic-monotile-exists/ https://aperiodical.com/2023/03/an-aperiodic-monotile-exists...
- johndough 4y agoProject website with web demo https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/ Direct link to PDF on ArXiv (89 pages) https://arxiv.org/pdf/2303.10798.pdf https://arxiv.org/pdf/2303.10798.pdf
- snowram 4y agoSounds like the start of BLIT by David Langford.
- shireboy 4y agoI'm confused - I see multiple repeating patterns. three light blue hats in triangle around dark blue. grey boomerang pattern. two white tiles with same rotation and layout. I'm sure I misunderstand what is meant by "pattern that never repeats", but please dumb this down for me.
- jsd1982 4y agoI was about to ask the same question but it appears the repeats are not exactly identical at the edges. Still, can one prove this is aperiodic from geometry alone? It seems rather difficult to actually prove that fact. Feels intuitive that there must be a period somewhere, however large it may be, on the infinite 2D plane.
- HelloNurse 4y agoFor most sets of shapes a periodic tiling is possible, but by no means guaranteed. For example, consider rectangles with sides of 1 and 3 units: they can cover the plane periodically (e.g. in a simple rectangular grid), but also aperiodically, because you can form a square grid of square 3 by 3 units "metatiles", each encoding one bit of information in the vertical or horizontal orientation of the narrow rectangles; then it's easy to break symmetry by orienting metatiles so that for all integers m and n some metatile differs from the metatile m rows and n columns away, so the period cannot be m rows and n columns.
- didericis 4y agoI’m assuming the infinite sequence of segments along a straight vertical line at each “x-coordinate” (not sure how to say this, but you can see vertical lines made of blocks, mean those) were proved to be unique and non repeating
- mbi 4y ago"An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches" [1] So, it has "islands" of repenting combinations of tiles, but these islands do not repeat / translate in a regular way. 1: https://www.wikiwand.com/en/Aperiodic_tiling https://www.wikiwand.com/en/Aperiodic_tiling
- kzrdude 4y agoPenrose tilings have 5- or 10-fold symmetry right, what does this have? Maybe triangular symmetry? In their coloring I see lots of three-studded shuriken-like shapes.
- OscarCunningham 4y agoThis colouring makes the 3-fold symmetry more visible: https://mathstodon.xyz/@Danpiker/110062396666001681 https://mathstodon.xyz/@Danpiker/110062396666001681
- hgsgm 4y agoIt's obviously hexagonal (which includes triangular) just by looking at the tiles and the angles of the edges.
- mg 4y agoThey provide this demo: https://cs.uwaterloo.ca/~csk/hat/app.html https://cs.uwaterloo.ca/~csk/hat/app.html I just wished one could turn off the colors. The colors really distract me from trying to see the patterns the shape itself creates. For me, the beauty here is that each piece is exactly the same. Colorizing them differently takes away from that.
- sjaak 4y agoTry this: main { filter: saturate(7) grayscale(10) contrast(3); }
- gwbas1c 4y ago> I just wished one could turn off the colors. It's not too hard to scrape the JavaScript out of the page. You could figure out where they set the colors and change that part. I also wonder if you can do that on https://mathigon.org/polypad/8kVqVH2Mor6JTQ https://mathigon.org/polypad/8kVqVH2Mor6JTQ There's also https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/, but sadly I didn't see anything like a github link to the above demo.
- dist-epoch 4y ago* for some definition of "never repeat". Most people would call that pattern obviously repeating (in the shape itself).
- bookofjoe 4y agohttps://archive.ph/RW7Wy https://archive.ph/RW7Wy
- aliljet 4y agoWhat a fantastically enjoyable read. I really want to understand how this question arises and what's actually being tested and invented in coming to the solution to this. And, side note, this is absolutely going to be the strategy I use to paint one of my office walls.
- zokier 4y agoI notice that each tile has 5 or 6 neighbors. This reminds me of "football" tiling[1] so I wonder how would this hat tiling look on non-euclidean geometry, e.g. on a spehere [1] https://commons.m.wikimedia.org/wiki/File:Comparison_of_truncated_icosahedron_and_soccer_ball.png https://commons.m.wikimedia.org/wiki/File:Comparison_of_trun...
- beeforpork 4y agoDefinitely nice for bathroom tiles! Staring at the wall while doing your business and failing to find a repeating pattern -- wonderful!
- bell-cot 4y agoIf I recall (previously-submitted article - https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/ https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/ ), the shape has to be flipped upside down some fraction of the time. So you'd either need two tile shapes for a real bathroom, or your tiles would be a compromise between "both faces are easily cleaned" and "both faces stick firmly to the mortar".
- gfd 4y agoSounds like a pretty tough job due to the lack of pattern. Either you have to follow a template or you run the risk of randomly tiling something that can't actually be extended further.
- dclowd9901 4y agoThe tiles seem like they can only join at one spot to one spot so I don’t think it would even be possible to lay them incorrectly unless you straight were jamming incorrect sides together.
- zokier 4y ago
- MontagFTB 4y agoMy kids and I geeked out over the Veritasium video on Penrose tiles (https://www.youtube.com/watch?v=48sCx-wBs34 https://www.youtube.com/watch?v=48sCx-wBs34). It is pretty exciting to see approachable math like this being discovered before our eyes.
- andrethegiant 4y agoOh shit new shape just dropped
- OJFord 4y agoJust the one? I have no idea how this is described mathematically, but just looking at the image in the article, the shape spans three hexagons, comprising 2/6 sectors of two of them and 4/6 of the third. I have no idea what I'm talking about, but it seems like it 'ought' to scale to larger (or at least some larger) polygons, or number of them spanned, even excluding trivial multiples (or 6/6 covered ones inserted in the middle) that are effectively the same shape.
- Someone 4y agoThere probably are countless others, but this is the first that we know of. And I wouldn’t know whether trivial multiples that still tile the plane non-periodically exist. Once you pick a multiple, even the claim that any of these basic structures in the plane is part of the multiple you picked doesn’t seem to have an obvious, trivial (1) proof to me, let alone the additional requirement that you can find non-overlapping ones. (1) I’m trying, likely unsuccessfully, to dodge the problem of triviality in mathematics (https://en.wikipedia.org/wiki/Triviality_(mathematics) https://en.wikipedia.org/wiki/Triviality_(mathematics)) here
- odgaus 4y agoOn their project page [1] they even mention that there is a (infinite) family of shapes >> The hat is one member of a continuous family of shapes that are all aperiodic, and that all tile the plane in the same way. [1] https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/
- penteract 4y agoIn the paper(linked in other comments), they say it's part of a family of such tiles which can be generated by changing some of the edge lengths.
- dclowd9901 4y agoFrom a maths standpoint, I’m curious what the analogy to numbers is. Would this tile be like an irrational number? A prime number? But in 2d space?
- DoctorMckay101 4y agoCowntdown until either Numberphile or Matt Parker does a video on this. starting now
- MC_10 4y agoPreviously on HN: https://news.ycombinator.com/item?id=35242458 https://news.ycombinator.com/item?id=35242458
- DFHippie 4y agoSomebody needs to manufacture a cookie cutter in this shape.
- satvikpendem 4y agoThere's a great Veritasium video about aperiodic tiling: https://www.youtube.com/watch?v=48sCx-wBs34 https://www.youtube.com/watch?v=48sCx-wBs34
- magicalhippo 4y agoEver since we went hunting for tiles for our first remodeling, I've been thinking about why not Wang tiles[1][2] were available. I mean obviously it'd be too much cost and hassle, since you need at least 5 different tiles to tile a plane, but I'm still curious how it would actually turn out on a real floor or wall, with an interesting pattern on the tiles. While you need multiple Wang tiles, at least they can be square rather than a rather awkward polygonal shape. So there's that... [1]: https://grahamshawcross.com/2012/10/12/wang-tiles-and-aperiodic-tiling/ https://grahamshawcross.com/2012/10/12/wang-tiles-and-aperio... [2]: https://en.wikipedia.org/wiki/Wang_tile https://en.wikipedia.org/wiki/Wang_tile
- cromulent 4y agoThey did a mall in Helsinki with Penrose tiles, I really like it. http://www.neverendingbooks.org/penrose-tiles-in-helsinki http://www.neverendingbooks.org/penrose-tiles-in-helsinki
- throwawaymaths 4y agoThe Salesforce center in SF looks like it's Penrose tiles, but I suspect they are cheating and are using a large segment repeated on each section of the skirt.
- RC_ITR 4y agoMaybe, but Penrose himself was involved, so I'd be very bummed if that were true. SAN FRANCISCO--(BUSINESS WIRE)--The Transbay Joint Powers Authority (TJPA) has received approval from Dr. Roger Penrose, the eminent British mathematical physicist, to incorporate his groundbreaking geometrical pattern in the design of the exterior walls of the future Transbay Transit Center (TTC) designed by Pelli Clarke Pelli Architects (PCPA). Dr. Penrose and PCPA are working in tandem to incorporate Dr. Penrose’s elegant design, known as the Penrose Rhombus Tiling, in the skin of the TTC. The design is remarkably simple but unique because it can be extended infinitely without repeating itself. The Penrose system is ideal for the perforations in the metal panels that will form the curved exterior of the Transit Center. https://www.businesswire.com/news/home/20130711006350/en/Roger-Penrose-Puts-Prints-on-San-Francisco-Transit-Center https://www.businesswire.com/news/home/20130711006350/en/Rog...
- 6nf 4y agoFlipping the shape is cheating imo. Might as well use Penrose for any actual tile work.
- jschveibinz 4y agoI’m not a mathematician, but it’s interesting to think about this as a projection onto 2d. What can be said about the multi-dimensional shape that creates this projection, or even if that is possible?
- ReaderView 4y ago[dead]
- brobdingnagians 4y agoOpportunity for a startup to start selling these, however niche that might be...
- hgsgm 4y agoLook for it at Cherry Arbor Design https://cherryarbordesign.com/collections/all https://cherryarbordesign.com/collections/all
- nashashmi 4y agosomewhere in this arrangement of tiles is a picture of the world. --veritasium
- hgsgm 4y agoIs the proven? Aperiodic is not the same as "full measure". 101001000100001... is aperiodic but doesn't contain every finite string. But you could say that becaus it contains an infinite set of distinct finite substrings, it can be put in bijection with any countable set of objects. That's not a "picture" in common parlance, via any sort of structured encoding, it's just an index.
- youssefabdelm 4y agoI wish it were a little more random in a sense... just enough so that the brain doesn't get "bored" of the evolution of the pattern, but not too much randomness that the randomness itself becomes like white noise (yet another pattern the brain can get "bored" of) Would be extremely curious if patterns like the one described exist in math.
- mckeed 4y agoI feel Penrose P2 tiles are better in that sense. I think you could craft something with that kind of "fractal interestingness" by using color to emphasize the larger regular patterns that can occur in a Penrose tiling. https://en.wikipedia.org/wiki/Penrose_tiling#Kite_and_dart_tiling_(P2) https://en.wikipedia.org/wiki/Penrose_tiling#Kite_and_dart_t...
- xhkkffbf 4y agoA wall? How about a floor? I wanted to cover my kitchen in Penrose tiles but they didn't seem to be available. Anyone know where to get some?
- sargstuff 4y agoagonizingly slow method: 3d print the appropriate amount of tiles (or at least a form mold for the tile clay)
- rascul 4y agoMight be interesting to consider if/when such tiles are available for purchase.
- bitterlesson 4y agoIf you have access to a laser cutter, you can make them out of wood or acrylic. You may find a laser cutter at your local library or maker space. I'd be happy to make tiles for you at the cost of materials and shipping.
- swayvil 4y agoIt's based on a sorta chunkified kisrhombille tiling (which is pretty sane). Which is based on 1-2-sqrt3 triangles. (Which are deeply humdrum). So we have a serious "infinite chaos out of plain order" situation here. Which I call impressive. We have like 10 different chaoses, depending on how you do your first tile. What would a superposition look like? And it's pretty easy to organize, given that it's based on the chunkykisrhombille. Hmmm. What powers would it give us, bigstructurewise?
- swayvil 4y agoWait I fucked that up. There aren't 10 chaoses.
- frankus 4y agoIs the similarity to a Dragon Curve (https://en.wikipedia.org/wiki/Dragon_curve https://en.wikipedia.org/wiki/Dragon_curve) a coincidence? This post (https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/) mentions a substitution system, which makes me think there might be a connection.
- teraflop 4y agoAs far as I know, the resemblance is superficial. (For one thing, the standard dragon curve is based on 90° angles, and this tile has a mixture of 90° and 120° angles, giving it a 6-fold pseudo-symmetry.) There are many other non-periodic or aperiodic tilings that are based on substitution rules, and many of them look totally different: https://tilings.math.uni-bielefeld.de/substitution/penrose-rhomb/ https://tilings.math.uni-bielefeld.de/substitution/penrose-r... https://tilings.math.uni-bielefeld.de/substitution/fibonacci-times-fibonacci-variant/ https://tilings.math.uni-bielefeld.de/substitution/fibonacci... https://tilings.math.uni-bielefeld.de/substitution/semi-detached-house/ https://tilings.math.uni-bielefeld.de/substitution/semi-deta... In addition, the dragon curve is a fractal -- mathematically, it's defined as the limit that the substitution process converges to as the details get "infinitely small", which means that in a sense, the true dragon curve (as opposed to the approximation that you can draw on a computer) has a boundary with no straight line segments at all. On the other hand, an aperiodic tiling is composed of finite, fixed-size tiles that extend outwards to infinity. Funnily enough, the dragon curve is a space-filling curve that tiles the plane periodically.
- uptownfunk 4y agohttps://archive.is/RW7Wy https://archive.is/RW7Wy
- ubj 4y ago> Until now, it wasn’t even clear whether such a single shape, known as an einstein (from the German “ein stein” or “one stone”), could even exist. The actual topic of the article was impressive, but this little fact about the meaning of "ein stein" was pretty interesting as well. TIL.
- chaxor 4y agoUseful to remember when someone says you're 'dumb as a rock'
- wussboy 4y agoI suppose you'd need to respond with, "Which rock? Specifically."
- wongarsu 4y agoThen you might also like Spielberg being German for "play mountain" or "game mountain", Adelson being German for "son of nobility" (though the son ending is more common in Nordic countries, the meaning is likely the same), Zuckerberg being German for "sugar mountain", Rosenberg being German for "rose mountain" or Friedman being old German for "peaceful man" or "protecting man".
- xdennis 4y agoAlso, it's pronounced neither "steen" nor "stayn", but "shtayn" or /ʃtaɪn/ in IPA. (I was once hearing someone talk about privacy and how people like Tsucabuc are destroying it. I never heard about him but apparently he is one of the owners of a large social media company. Then he mentioned Facebook and I realized he was pronouncing Zuckerberg in German.)
- hooverd 4y agoFloor installers HATE this one simple shape!
- litoE 4y agoI dont't get it. In the picture, each tile is composed of 8 identical quadrilaterals. So why don't these quadrilaterals constitute a simpler shape that can tile a wall and never repeat?
- twanvl 4y agoThis tile forces a pattern that does not repeat. If you use the simpler shape you can tile a wall such that it never repeats, but you can also make a repeating pattern.
- plopilop 4y agoThe key point (often missed by these articles) is that aperiodic tilings do not have a (infinite) periodic pattern. This means that you cannot draw a shape on these tiles and say: "the tiling is based on infinite repetitions of this shape, and only this shape". Of course individual tiles will repeat, but never in an infinite periodic pattern. Edit: a novelty of this paper is that their shape is "truly" aperiodic, which means no matter how hard you try, you will end up with aperiodic tiling. Existing one-shape aperiodic tilings had to add constraints on how to put two shapes next to each other to ensure aperiodicity.
- notfed 4y ago"Of course individual tiles will repeat" What does this mean? How can an individual tile "repeat"?
- plopilop 4y agoI meant "a given orientation of the tile will be present many times (or infinitely many)". It's very probable (I did not read the paper, only their website page) that the tile only occupies a finite amount of orientations in the tiling and therefore at least (and probably more if not all) one orientation will also be present an infinite amount of times. However this does not imply periodicity of the tiling.
- teawrecks 4y ago
- paulpauper 4y agoThis seems like the sort of thing Terrance Tao or a computer should have solved long ago.
- swayvil 4y agoThat is a seriously chewed cookie.
- tagami 4y agoOn an infinite plane, math is telling us that there is no pattern. Is this correct?
- swayvil 4y agoThis geometry is a cousin of that geometry http://www.fleen.org/generative_art_project/i0_quartersize.png http://www.fleen.org/generative_art_project/i0_quartersize.p...
- bassrattle 4y agoI'd really like to see this applied to 3D world modeling. If a landscape were tiled with a textured material in this way, perhaps it would look more natural.
- patrickwalton 4y agoI would love this, but also it would be way more expensive to tile, because you can't manufacture a consistent sheet of tiles. No two sheets would be alike!
- sys42590 4y agoSo it would be possible to make a new tiling for Tatham's Loopy puzzle [0] that would surely look nice. [0]: https://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/loopy.html https://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/loop...
- sys42590 4y agoYay: https://git.tartarus.org/?p=simon/puzzles.git;a=commit;h=8d6647548f7d0051221374ad6eb2b6dd32e2a3ed https://git.tartarus.org/?p=simon/puzzles.git;a=commit;h=8d6...
- 1MachineElf 4y agoI wonder what applications there are for this in video games. Many games that attempt to show the outside world often suffer from repeating patterns in things like terrain, which would never happen in real life. Everything from 2D isometric games like Command & Conquer to 3D open worlds like Skyrim have this problem. Could that problem be solved by tile shapes that never produce repeating patterns?
- cadooo 4y agoMy immediate thought was how soon do I see this in a board game. Hexs are often used to create the game board. This could add a lot of variability to board setup.
- wetmore 4y agoPreventing texture repetition is definitely one area you'll see techniques like this, e.g. via Wang tilings. Here is another example: https://iquilezles.org/articles/texturerepetition/ https://iquilezles.org/articles/texturerepetition/
- 1MachineElf 4y agoFascinating article and code examples. Thanks for the link.
- TinkersW 4y agoThere are already techniques for removing the repeating textures(sometimes this is called texture bombing). Maybe this can be used to improve them, but only if it can be cheaply calculated on the fly.
- robinsonb5 4y agoSimilarly, I found myself wondering about applications in halftone patterns for printing.
- EGreg 4y agoSo the Wang hypothesis was disproven, then?
- bitsinthesky 4y agoNow, how many colors would you need so that no two adjacent regions have the same color?
- thefringthing 4y agoAll four, since it's fairly easy to find an odd wheel in the tiling.
- dmtroyer 4y agoKind of looks like overlapping t-shirts.
- justinator 4y agoSomething tells me if one was to look closely you'd find this tile in Clark Richert's work, probably while he was living in an artist commune in South Colorado in the 60's (like they did with the Penrose Tile, which he got sued for - and won, since he showed prior art). https://www.google.com/search?q=%22Clark+Richert%22+Art&tbm=isch https://www.google.com/search?q=%22Clark+Richert%22+Art&tbm=...
- pmayrgundter 4y agoThat suggests that the layout at some radius R in the distance is unpredictable without "running" it, in the Wolfram computational-irreducible sense. They say they have "a new kind of geometric incommensurability argument", and there are many statements about the related undecidability of related tiling classes.. but not really grokking this. Anyone know?
- cwmoore 4y agoAre there numeric indexes for locations in an aperiodic monotile covering a plane such as there are for the xyz in a tiled web map?
- bongoman37 4y ago[dead]
- funny_falcon 4y agoBut there are two shapes: "enshtein" and its reflection.
- sargstuff 4y agoAh, a proof for updating the busy beaver / Turing machine to include GPT: the busy tiler -- 'walk this way, talk this way'[1]. Perhaps also improving computational users health with a python corollary[0]. [0] : https://www.cnn.com/videos/health/2023/01/09/monty-python-silly-walk-study-lbb-orig-nb.cnn https://www.cnn.com/videos/health/2023/01/09/monty-python-si... [1] : https://en.wikipedia.org/wiki/Walk_This_Way https://en.wikipedia.org/wiki/Walk_This_Way
- rajnathani 4y agoUnbelievable work, including the existing work with Wang tiles and others mentioned in adjacent HN comments. Side: This is an amazing tattoo idea.