Mathematicians discover shape that can tile a wall and never repeat

Project website with web demo <a href="https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;" rel="nofollow">https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;</a><p>Direct link to PDF on ArXiv (89 pages) <a href="https:&#x2F;&#x2F;arxiv.org&#x2F;pdf&#x2F;2303.10798.pdf" rel="nofollow">https:&#x2F;&#x2F;arxiv.org&#x2F;pdf&#x2F;2303.10798.pdf</a>

Ever since we went hunting for tiles for our first remodeling, I&#x27;ve been thinking about why not Wang tiles[1][2] were available.<p>I mean obviously it&#x27;d be too much cost and hassle, since you need at least 5 different tiles to tile a plane, but I&#x27;m still curious how it would actually turn out on a real floor or wall, with an interesting pattern on the tiles.<p>While you need multiple Wang tiles, at least they can be square rather than a rather awkward polygonal shape. So there&#x27;s that...<p>[1]: <a href="https:&#x2F;&#x2F;grahamshawcross.com&#x2F;2012&#x2F;10&#x2F;12&#x2F;wang-tiles-and-aperiodic-tiling&#x2F;" rel="nofollow">https:&#x2F;&#x2F;grahamshawcross.com&#x2F;2012&#x2F;10&#x2F;12&#x2F;wang-tiles-and-aperio...</a><p>[2]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wang_tile" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Wang_tile</a>

&gt; 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.<p>The actual topic of the article was impressive, but this little fact about the meaning of &quot;ein stein&quot; was pretty interesting as well. TIL.

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

They provide this demo:<p><a href="https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;app.html" rel="nofollow">https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;app.html</a><p>I just wished one could turn off the colors.<p>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.

<a href="https:&#x2F;&#x2F;archive.is&#x2F;iqBHP" rel="nofollow">https:&#x2F;&#x2F;archive.is&#x2F;iqBHP</a>

Previously on HN: <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=35242458" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=35242458</a>

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.

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&#x2F;6 sectors of two of them and 4&#x2F;6 of the third. I have no idea what I&#x27;m talking about, but it seems like it &#x27;ought&#x27; to scale to larger (or at least some larger) polygons, or number of them spanned, even excluding trivial multiples (or 6&#x2F;6 covered ones inserted in the middle) that are effectively the same shape.

It&#x27;s based on a sorta chunkified kisrhombille tiling (which is pretty sane). Which is based on 1-2-sqrt3 triangles. (Which are deeply humdrum).<p>So we have a serious &quot;infinite chaos out of plain order&quot; situation here. Which I call impressive.<p>We have like 10 different chaoses, depending on how you do your first tile. What would a superposition look like?<p>And it&#x27;s pretty easy to organize, given that it&#x27;s based on the chunkykisrhombille.<p>Hmmm. What powers would it give us, bigstructurewise?

My kids and I geeked out over the Veritasium video on Penrose tiles (<a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=48sCx-wBs34">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=48sCx-wBs34</a>). 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&#x27;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?

There&#x27;s a great Veritasium video about aperiodic tiling: <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=48sCx-wBs34">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=48sCx-wBs34</a>

Something tells me if one was to look closely you&#x27;d find this tile in Clark Richert&#x27;s work, probably while he was living in an artist commune in South Colorado in the 60&#x27;s (like they did with the Penrose Tile, which he got sued for - and won, since he showed prior art).<p><a href="https:&#x2F;&#x2F;www.google.com&#x2F;search?q=%22Clark+Richert%22+Art&amp;tbm=isch" rel="nofollow">https:&#x2F;&#x2F;www.google.com&#x2F;search?q=%22Clark+Richert%22+Art&amp;tbm=...</a>

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 &amp; Conquer to 3D open worlds like Skyrim have this problem. Could that problem be solved by tile shapes that never produce repeating patterns?

I notice that each tile has 5 or 6 neighbors. This reminds me of &quot;football&quot; tiling[1] so I wonder how would this hat tiling look on non-euclidean geometry, e.g. on a spehere<p>[1] <a href="https:&#x2F;&#x2F;commons.m.wikimedia.org&#x2F;wiki&#x2F;File:Comparison_of_truncated_icosahedron_and_soccer_ball.png" rel="nofollow">https:&#x2F;&#x2F;commons.m.wikimedia.org&#x2F;wiki&#x2F;File:Comparison_of_trun...</a>

Oh shit new shape just dropped

I&#x27;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&#x27;m sure I misunderstand what is meant by &quot;pattern that never repeats&quot;, but please dumb this down for me.

<a href="https:&#x2F;&#x2F;archive.ph&#x2F;RW7Wy" rel="nofollow">https:&#x2F;&#x2F;archive.ph&#x2F;RW7Wy</a>

Can someone ELI5 why this is different from Penrose tiling?

Ah, a proof for updating the busy beaver &#x2F; Turing machine to include GPT: the busy tiler -- &#x27;walk this way, talk this way&#x27;[1]. Perhaps also improving computational users health with a python corollary[0].<p>[0] : <a href="https:&#x2F;&#x2F;www.cnn.com&#x2F;videos&#x2F;health&#x2F;2023&#x2F;01&#x2F;09&#x2F;monty-python-silly-walk-study-lbb-orig-nb.cnn" rel="nofollow">https:&#x2F;&#x2F;www.cnn.com&#x2F;videos&#x2F;health&#x2F;2023&#x2F;01&#x2F;09&#x2F;monty-python-si...</a><p>[1] : <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Walk_This_Way" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Walk_This_Way</a>

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 (<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dragon_curve" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dragon_curve</a>) a coincidence? This post (<a href="https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;" rel="nofollow">https:&#x2F;&#x2F;cs.uwaterloo.ca&#x2F;~csk&#x2F;hat&#x2F;</a>) mentions a substitution system, which makes me think there might be a connection.

That suggests that the layout at some radius R in the distance is unpredictable without &quot;running&quot; it, in the Wolfram computational-irreducible sense.<p>They say they have &quot;a new kind of geometric incommensurability argument&quot;, and there are many statements about the related undecidability of related tiling classes.. but not really grokking this.<p>Anyone know?

I wish it were a little more random in a sense... just enough so that the brain doesn&#x27;t get &quot;bored&quot; 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 &quot;bored&quot; of)<p>Would be extremely curious if patterns like the one described exist in math.

So it would be possible to make a new tiling for Tatham&#x27;s Loopy puzzle [0] that would surely look nice.<p>[0]: <a href="https:&#x2F;&#x2F;www.chiark.greenend.org.uk&#x2F;~sgtatham&#x2F;puzzles&#x2F;js&#x2F;loopy.html" rel="nofollow">https:&#x2F;&#x2F;www.chiark.greenend.org.uk&#x2F;~sgtatham&#x2F;puzzles&#x2F;js&#x2F;loop...</a>

<a href="https:&#x2F;&#x2F;archive.is&#x2F;RW7Wy" rel="nofollow">https:&#x2F;&#x2F;archive.is&#x2F;RW7Wy</a>

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

This geometry is a cousin of that geometry<p><a href="http:&#x2F;&#x2F;www.fleen.org&#x2F;generative_art_project&#x2F;i0_quartersize.png" rel="nofollow">http:&#x2F;&#x2F;www.fleen.org&#x2F;generative_art_project&#x2F;i0_quartersize.p...</a>

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

I dont&#x27;t get it. In the picture, each tile is composed of 8 identical quadrilaterals. So why don&#x27;t these quadrilaterals constitute a simpler shape that can tile a wall and never repeat?

Unbelievable work, including the existing work with Wang tiles and others mentioned in adjacent HN comments.<p>Side: This is an amazing tattoo idea.

I&#x27;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&#x27;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&#x27;t seem to be available. Anyone know where to get some?

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&#x2F;when such tiles are available for purchase.

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.

But there are two shapes: &quot;enshtein&quot; 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 &quot;never repeat&quot;.<p>Most people would call that pattern obviously repeating (in the shape itself).

somewhere in this arrangement of tiles is a picture of the world.<p>--veritasium