Mathematicians discover shape that can tile a wall and never repeat | Hacker News Reader

Mathematicians discover shape that can tile a wall and never repeat

newscientist.com

488 pointsiamben156 comments

Project website with web demo https://cs.uwaterloo.ca/~csk/hat/

Direct link to PDF on ArXiv (89 pages) https://arxiv.org/pdf/2303.10798.pdf

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...

[1]: https://grahamshawcross.com/2012/10/12/wang-tiles-and-aperio...

[2]: https://en.wikipedia.org/wiki/Wang_tile

They did a mall in Helsinki with Penrose tiles, I really like it.

http://www.neverendingbooks.org/penrose-tiles-in-helsinki

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.

https://www.businesswire.com/news/home/20130711006350/en/Rog...

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

Isn't the point of Penrose tiles that it's made of 2 shapes (i.e. you only need two arbitrarily movable stamps to make it work)?

Why not? If it's laser cut, then it's almost no additional cost to make each sheet different. It's all software.

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

Out of curiosity, is it idiomatic to call a street next to a shopping center a mall? Ignorant Finn here.

No, probably my age and background. Street with shops and no cars.

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.

"Mall" can refer to a pedestrian area within a city more generally; shops are optional, e.g.,

https://nps.gov/nama/index.htm

Also, sometimes cars,

https://www.royalparks.org.uk/parks/st-jamess-park/things-to...

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”.

Or, on a much smaller scale...

http://www.tessellations.org/real-materials-tessellations-16...

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.

Re : cool tiling options

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.

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):

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...

seems as if this could be solved by using a different palette. The red sticks out cuz it’s deep dark red more than cuz it’s dominoes. Maybe.

Maybe, I think in general though, this falls into uncanny valley territory.

Humans like patterns or randomness, we sort of hate quasi-versions of either (See: our disdain for blurry pictures).

I love the idea, maybe not in wiki colors but with some stylization. One concern thought, how can I know how many tiles of each type I will need?

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.

Not Wang tiles, but a quasiperiodic tiling discovered by Robert Ammann: https://www.math.ucdavis.edu/~greg/floors.html

> 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.

Useful to remember when someone says you're 'dumb as a rock'

I suppose you'd need to respond with, "Which rock? Specifically."

Then 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".

Also, 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.)

Definitely nice for bathroom tiles! Staring at the wall while doing your business and failing to find a repeating pattern -- wonderful!

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".

> The ratio of unreflected to reflected tiles is φ^4 : 1

I just love this

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.

Wow that’s really something, thanks for the additional info

Who needs a never-ending smartphone feed if you can have a never-ending tile pattern to gaze upon!

Cool, but for some reason i find it pretty hard to look at?

stop looking at it :)

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:

https://tilings.math.uni-bielefeld.de/

examples:

https://tilings.math.uni-bielefeld.de/substitution/domino-va...

https://tilings.math.uni-bielefeld.de/substitution/tetris/

Thanks for posting this. It should be higher.

I can't believe NewScientist left that out of their headline.

>Bielefeld university

I'm not clicking that, I don't trust I'll be able to come back.

For the downvoters, I'm pretty sure this is supposed to be a veiled reference to the https://en.wikipedia.org/wiki/Bielefeld_conspiracy

I've heard of this injoke a lot, but it doesn't seem to have any real core of fun to it, it's just interesting because it's an injoke?

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.

Forcing aperiodicity makes the installation idiot-proof though.

It's been a looooong time ....

But I could see this making a "psychedelic" experience a bit more interesting.

They provide this demo:

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.

Try this:

main { filter: saturate(7) grayscale(10) contrast(3); }

> 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

There's also https://cs.uwaterloo.ca/~csk/hat/, but sadly I didn't see anything like a github link to the above demo.

https://archive.is/iqBHP

Previously on HN: https://news.ycombinator.com/item?id=35242458

Also: https://news.ycombinator.com/item?id=35264965

And: https://news.ycombinator.com/item?id=35265569

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.

This colouring makes the 3-fold symmetry more visible: https://mathstodon.xyz/@Danpiker/110062396666001681

It's obviously hexagonal (which includes triangular) just by looking at the tiles and the angles of the edges.

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.

(1) I’m trying, likely unsuccessfully, to dodge the problem of triviality in mathematics (https://en.wikipedia.org/wiki/Triviality_(mathematics)) here

On 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/

In 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.

From 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?

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.

It'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?

Wait I fucked that up. There aren't 10 chaoses.

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.

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’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?

That was my first thought too.

Very interesting. There are specific regions that repeat, but the overall image does not.

This tiling is hexagons painted differently, and hexagons are cubes projected onto a plane.

There's a great Veritasium video about aperiodic tiling: https://www.youtube.com/watch?v=48sCx-wBs34

Something 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=...

Somebody needs to manufacture a cookie cutter in this shape.

I 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?

My 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.

Preventing 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/

Fascinating article and code examples. Thanks for the link.

There 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.

Similarly, I found myself wondering about applications in halftone patterns for printing.

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

[1] https://commons.m.wikimedia.org/wiki/File:Comparison_of_trun...

Oh shit new shape just dropped

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

"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

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.

The claim is that this tesselation is aperiodic, meaning that the entire pattern does not have translational symmetry.

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.

See: https://personal.math.ubc.ca/~cass/courses/m308-02b/projects...

It only repeats for a little bit.

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.

Thank you for explaining "aperiodic" in a way that makes sense to non-geometers!

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.

https://archive.ph/RW7Wy

Can someone ELI5 why this is different from Penrose tiling?

As I understand it: they found a single shape. Where existing Penrose tilings were composed of two (or more) shapes.

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.

...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.

Rigid transformations of a single shape https://en.wikipedia.org/wiki/Rigid_transformation

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.

It is just a single tile type, not two

The two Penrose tiles are also an affine transformation of each other.

Whereas in this new tiling there's a single shape and its rigid transformations.

I don't think they can be if you want to enforce the matching rules using the tile's shape.

> matching rules using the tile's shape.

That excludes affine transformation.

That's covered in the article.

It has a paywall.

But almost every HN post to pay-walled content includes an archive.is link to bypass the paywall.

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.

https://archive.is/RW7Wy

Ah, 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-si...

[1] : https://en.wikipedia.org/wiki/Walk_This_Way

Flipping the shape is cheating imo. Might as well use Penrose for any actual tile work.

Cowntdown until either Numberphile or Matt Parker does a video on this. starting now

Floor installers HATE this one simple shape!

Is the similarity to a Dragon Curve (https://en.wikipedia.org/wiki/Dragon_curve) a coincidence? This post (https://cs.uwaterloo.ca/~csk/hat/) mentions a substitution system, which makes me think there might be a connection.

As 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-r...

https://tilings.math.uni-bielefeld.de/substitution/fibonacci...

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.

That 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?

I 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.

I 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_t...

So 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/loop...

Yay: https://git.tartarus.org/?p=simon/puzzles.git;a=commit;h=8d6...

https://archive.is/RW7Wy

Now, how many colors would you need so that no two adjacent regions have the same color?

All four, since it's fairly easy to find an odd wheel in the tiling.

This geometry is a cousin of that geometry

http://www.fleen.org/generative_art_project/i0_quartersize.p...

Opportunity for a startup to start selling these, however niche that might be...

Look for it at Cherry Arbor Design https://cherryarbordesign.com/collections/all

I 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?

This 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.

The 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.

"Of course individual tiles will repeat"

What does this mean? How can an individual tile "repeat"?

I 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.

Someone below mentioned that the important part isn't that it tiles aperiodically, but that it ONLY tiles aperiodically. There are simpler shapes that will tile both aperiodically and periodically.

Unbelievable work, including the existing work with Wang tiles and others mentioned in adjacent HN comments.

Side: This is an amazing tattoo idea.

I'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.

I 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!

A 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?

agonizingly slow method: 3d print the appropriate amount of tiles (or at least a form mold for the tile clay)

Are there numeric indexes for locations in an aperiodic monotile covering a plane such as there are for the xyz in a tiled web map?

This seems like the sort of thing Terrance Tao or a computer should have solved long ago.

Might be interesting to consider if/when such tiles are available for purchase.

If 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.

On an infinite plane, math is telling us that there is no pattern. Is this correct?

Roughly it looks like a T-shirt. One with a jagged bottom. Jagged T-shirt tile.

Yes, this article goes more into the attire aspects:

https://aperiodical.com/2023/03/an-aperiodic-monotile-exists...

But there are two shapes: "enshtein" and its reflection.

In the second image, the tiles look like West Virginia.

Sounds like the start of BLIT by David Langford.

So the Wang hypothesis was disproven, then?

Kind of looks like overlapping t-shirts.

That is a seriously chewed cookie.

* for some definition of "never repeat".

Most people would call that pattern obviously repeating (in the shape itself).

somewhere in this arrangement of tiles is a picture of the world.

--veritasium

Is 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.

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