
Self-tiling tile set - vinchuco
https://en.wikipedia.org/wiki/Self-tiling_tile_set
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reedlaw
I wonder if any of these tile sets could work in a Tetris-like game.

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BillTheCat
When you complete a tile shape using smaller tiles the game zooms out and the
shape you just completed is falling into a new self-tiling shape. _Inception
Noise_

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tunnuz
Very interesting. Is there a real-world application for this?

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teraflop
It reminds me of a technique for using recursive self-similar tiles to
generate pseudorandomly distributed points:
[http://johanneskopf.de/publications/blue_noise/](http://johanneskopf.de/publications/blue_noise/)

The idea is that you can model an infinite set of points generated by an
infinitely-deeply-nested tiling, and then efficiently sample from it at
arbitrary levels of detail. This lets you do things like zoom in and out while
retaining frame-to-frame coherence, and without having to explicitly store the
entire point set. (It's worth watching the full 5-minute video, which explains
it better than I can.)

The implementation in that paper used squares with labeled edges to tile the
plane non-periodically. This seems like it could be used to do something
similar, except with a periodic but irregular tiling. No idea if that would
have any practical benefits, but it's interesting, at least.

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anon4
Oh wow, I was looking for ways to do exactly this the other day.

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vishnuharidas
I'd rather get these tiles for paving in my house floor.

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amelius
These tiles have nice properties, but it seems the property that you can cover
an arbitrary shape with them is missing (even if allowing cropping of the
result).

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gjm11
If you allow cropping, you can always do this.

Suppose the tiles are T1, ..., Tn. The "self-tiling" property means you can
tile each Tk with smaller copies of T1, ..., Tn. Or, equivalently, you can
tile a larger copy of each Tk with copies of T1, ..., Tn.

So: pick one of the tiles. Tile it with smaller copies of the tiles. Tile each
of those with still-smaller copies. Tile each of _those_ with still-smaller
copies. Etc.

Now pick a region in the original tile that's the same shape as the thing
you're trying to cover. This process covers (the whole original tile, and
hence in particular) that region with finer and finer copies of the tiles.

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amelius
Yes, it seems simple, but I'm not sure if this proof is sufficiently rigorous
that it allows for mathematically pathological tiles.

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gjm11
I don't think it requires anything more than that at least one tile has
nonempty interior.

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maweki
The article is an interesting read. I wonder why it took so long to realize
there are longer loops.

I'd think, but this may well be hindsight speaking, about everything that
holds for tearm rewriting should hold for tiling.

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reilly3000
many thanks for sharing. This is rather amazing to me.

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brento
That feels a little like Inception
([https://en.wikipedia.org/wiki/Inception](https://en.wikipedia.org/wiki/Inception))
or the Infinity mirror
([https://en.wikipedia.org/wiki/Infinity_mirror](https://en.wikipedia.org/wiki/Infinity_mirror)).

