But 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.
It'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.
They 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.
I'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.
I 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.
For 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.
I’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
For example, you can't replace these tiles with just rectangles. For regular shapes you can, for example if you tile your bathroom with squares there will be a repeating 2x2 square pattern too. Similarly, hexagons and triangles can be replaced with rectangular tiles (where every tile is the same). You can't do this for this pattern.
Whatever the size of a pattern you find, you cannot tile the plane with translations (just translations) using that pattern: you need to rotate it. The definition is just that.
So, there are "patterns" (as a matter of fact, the elementary tile is repeated infinitely on the tiling) but you cannot fill the plane with mere translations of one.
Those configurations appear often, but they don't appear with a period. Look at a chessboard.. i can make it out of individual black and white tiles, or out of pre-assembled units of 1 white and one black tile (or a strip of length 4 or squares of size 4, etc). I can make a chess board by grouping the base units into a pattern, then only using that pattern. I can put that pattern on a wheel and roll it along a surface forever to get an infinite chessboard pattern.
There's not a way to do that sort of grouping for these tiles.
Compare pi. I can find the numbers representing my name in ascii an infinite number of times in the digits of pi, but if i find it once, there's no information in where to find it again, i can't just move forward n digits to find it, then another n digits to find it again, and ao on.
If you were to make two copies of the pattern and superimpose them on each other, they would never match perfectly no matter how you rotated them and shifted them around.
However, they might match in small patches, just never across the entire infinite plane.
wait, you must mean if you created a copy on top of another copy, they would match like that, but there's no combination of shifting, rotation, mirroring, etc that would also match? (unless it you shifted it back to the starting orientation)
cuz it would be mind blowing if you made a copy and it wasn't a copy... pauli whackamole exclusion tiling
It looks to me like it repeats in 1 direction pretty quickly, but in other directions doesn't repeat at all... At least, as far as I can tell from the samples given.
Penrose 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.
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.
What 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.
I 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
If I recall (previously-submitted article - 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".
Having a white tile and a blue tile would be fine, though. I'm really tempted by this, but I think I should start by tiling the garage floor or something rather than my whole living area.
Sounds 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.
The 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.
> First we produce a list of possible neighbours of the hat polykite in a tiling. There are 58 possible neighbours when we only require such a neighbour not to intersect the original polykite; these are shown in Figure B.1, with that original polykite shaded. The first 41 of these neighbours remain in consideration for the enumeration of 1-patches. The final 17 are immediately eliminated (in the order shown) because they cannot be extended to a tiling: either there is no possible neighbour that can contain the shaded kite (without resulting in an intersection, or a pair of tiles that were previously eliminated as possible neighbours), or we eliminated Y as a neighbour of X and so can also eliminate X as a neighbour of Y.
From the preprint. It is definitely possible to lay the tiles so that they do not tile anymore.
The unique feature of the new tile isn't that it will tile aperiodically. It's that it will only tile aperiodically. If you just want an aperiodic tiling, you can achieve it with 2x1 rectangles. There's a big list of such patterns here:
It is the earliest German bit of net culture to take an offramp from the information superhighway into regular culture - at least the earliest I know of. Maybe that gives it its staying power, humorous value aside.
My kids and I geeked out over the Veritasium video on Penrose tiles (https://www.youtube.com/watch?v=48sCx-wBs34). It is pretty exciting to see approachable math like this being discovered before our eyes.
Just 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.
There 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.
In 1D you can't have a single shape tile aperiodically, since there is no room for variation.
You could make a 2D diagram of the integers with they prime factorizations, which is aperiodic, but nearly periodic, but requires an infinite set of different "tiles".
Perhaps you could take an irrational or transcendebtal number, and take its multiples or powers mod 1, to get an aperiodic nearly periodic sequence.
Ever 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...
The 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.
Maybe, 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.
The reason why I'm skeptical is because it's cut out metal. Actual tiles would be easy to do it "correctly", but I don't imagine them I dunno, laser cutting? each skirt piece individually. I'd love to be proven wrong though
Yeah. I wonder how the tilesetters actually did the concrete tile-laying part. There are "quasiperiodic" Penrose tilings that are composed of fairly regular "macroblocks", but this Keskuskatu tiling looks pretty random to me. Did they just have huge printouts of the pattern? Did they develop an intuition of which tile goes where after a while? Were they happy to do something different for a chance, or annoyed by the convoluted task? :D
I think in the US that would colloquially be called an:
Outdoor mall
Mall alone being the 80s style indoor walkable variety. A 'strip mall' usually a single (often deformed to some degree) line of stores along a sidewalk next to a huge parking lot, also often with an island restaurant or small store that wants to stand out closer to the street edge of said lot.
Sometimes. “Mall” alone generally refers to a big indoor shopping space with many distinct stores; I'd probably call the outdoor version an... “outdoor mall”.
I don’t think this would be too much cost and hassle at all. People make all sorts of fancy tiles and patterns with them. Four colors as triangles is so simple I’m really surprised it doesn’t exist. Maybe a good niche online business for someone.
Purely from an aesthetics standpoint, I'm guessing its because they have a tendency to form quasi-repeating patterns and long rivers which are generally unsightly (which is subjective):
Yeah definitely not the colors used to highlight the pattern...
The way you construct a set of Wang tiles determines how many you'll need. The first link in my previous post shows how to construct them for various set sizes.
I’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?
161 comments
[ 2.9 ms ] story [ 143 ms ] threadKind of, though, right? One could also look at it as they've found two shapes that happen to be reflections of each other.
That excludes affine transformation.
https://aperiodical.com/2023/03/an-aperiodic-monotile-exists...
Direct link to PDF on ArXiv (89 pages) https://arxiv.org/pdf/2303.10798.pdf
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.
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.
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
So, there are "patterns" (as a matter of fact, the elementary tile is repeated infinitely on the tiling) but you cannot fill the plane with mere translations of one.
See: https://personal.math.ubc.ca/~cass/courses/m308-02b/projects...
There's not a way to do that sort of grouping for these tiles.
Compare pi. I can find the numbers representing my name in ascii an infinite number of times in the digits of pi, but if i find it once, there's no information in where to find it again, i can't just move forward n digits to find it, then another n digits to find it again, and ao on.
However, they might match in small patches, just never across the entire infinite plane.
cuz it would be mind blowing if you made a copy and it wasn't a copy... pauli whackamole exclusion tiling
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.
main { filter: saturate(7) grayscale(10) contrast(3); }
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
There's also https://cs.uwaterloo.ca/~csk/hat/, but sadly I didn't see anything like a github link to the above demo.
Most people would call that pattern obviously repeating (in the shape itself).
[1] https://commons.m.wikimedia.org/wiki/File:Comparison_of_trun...
I just love this
From the preprint. It is definitely possible to lay the tiles so that they do not tile anymore.
https://tilings.math.uni-bielefeld.de/
examples:
https://tilings.math.uni-bielefeld.de/substitution/domino-va...
https://tilings.math.uni-bielefeld.de/substitution/tetris/
I can't believe NewScientist left that out of their headline.
I'm not clicking that, I don't trust I'll be able to come back.
But I could see this making a "psychedelic" experience a bit more interesting.
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)) here
>> 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/
You could make a 2D diagram of the integers with they prime factorizations, which is aperiodic, but nearly periodic, but requires an infinite set of different "tiles".
Perhaps you could take an irrational or transcendebtal number, and take its multiples or powers mod 1, to get an aperiodic nearly periodic sequence.
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-aperio...
[2]: https://en.wikipedia.org/wiki/Wang_tile
http://www.neverendingbooks.org/penrose-tiles-in-helsinki
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/Rog...
Outdoor mall
Mall alone being the 80s style indoor walkable variety. A 'strip mall' usually a single (often deformed to some degree) line of stores along a sidewalk next to a huge parking lot, also often with an island restaurant or small store that wants to stand out closer to the street edge of said lot.
https://nps.gov/nama/index.htm
Also, sometimes cars,
https://www.royalparks.org.uk/parks/st-jamess-park/things-to...
http://www.tessellations.org/real-materials-tessellations-16...
There's girih tiles. It's like 5 tiles that you can do a bunch of stuff with. Invented by some based geomystics like 1000 years ago.
And there's kisrhombille. Lots of options there.
The long string of horizontal yellow diamonds and upside down blue "darts" in this image: https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/Wa...
Or the red "dominoes" in this image: https://grahamshawcross.files.wordpress.com/2012/10/13tiling...
Humans like patterns or randomness, we sort of hate quasi-versions of either (See: our disdain for blurry pictures).
The way you construct a set of Wang tiles determines how many you'll need. The first link in my previous post shows how to construct them for various set sizes.
Very interesting. There are specific regions that repeat, but the overall image does not.