Self-tiling tile set | Hacker News Reader

Self-tiling tile set

en.wikipedia.org

139 pointsvinchuco21 comments

I wonder if any of these tile sets could work in a Tetris-like game.

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

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

It reminds me of a technique for using recursive self-similar tiles to generate pseudorandomly distributed points: 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.

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

Bathrooms?

There need not be. 1+1=2 is true whether we are adding dollars or lines of code. Another answer is that I'm trying to make a puzzle with different solutions, all of which can't be solved simultaneously.

I never said that there must be an application, I was wondering whether there was one.

My first thought was for tiled textures, to make repetition less obvious

Wouldn't it be difficult to make a texture whose edges were contiguous with every possible permutation?

https://en.wikipedia.org/wiki/Digital_camouflage ?

I think that depends on the world you're trying to apply it to.

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

I was thinking the same. I've got a couple bathrooms that need updating.

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).

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.

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

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

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.

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

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

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