Post #2454891

2026-05-09 22:50 UTC

@simontatham@hachyderm.io @csk@mathstodon.xyz @pieter@mathstodon.xyz thanks for the link! I've read through the post and indeed you came up with the same idea of fractal coordinates but for a much harder case (I was applying it to simple squares-in-squares stuff). Advantages that I was interested in which don't apply to your case: you can generate arbitrary patches of the tiling incrementally as a user explores stuff, and you're guaranteed that no matter what path they take, things always match up when they return to somewhere they've been before. So you can do Minecraft-style incremental generation of a huge world and place stuff based on local decisions with logarithmic consultation of higher-level stuff being enough to guarantee aperiodicity, for example. Your idea for sampling from the correct distributions is genius, and it makes me want to try to put together a more "correctly distributed" version of my infinite path demo, although I'll have to think some more about the "correct" distributions of path types through an arbitrary square cutout sampled "uniformly" from all possible infinite Hamiltonian paths through a 1x1 infinite grid.

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@simontatham@hachyderm.io 2026-05-10 14:05

@tiotasram@kolektiva.social @csk@mathstodon.xyz @pieter@mathstodon.xyz generating incrementally as the user explores: yes, indeed, I've had that thought too. As you say, for _my_ ultimate application of generating a puzzle game, or the much more obvious one of just generating a patch of tiles for a pretty picture, there's no need to remember your state and extend it later, but you could perfectly well do that if you wanted to! On generating a tiling from the limiting distribution: since I wrote that, I've realised that phrase is ambiguous in some cases. If not all tile types have the same area, do you take the limiting distribution to be the distribution of _tiles_, or of _area_? Put another way, are we taking the limit as patch size → ∞ of the operation "pick a tile uniformly at random from this finite patch", or "pick a _point_ uniformly at random in the patch and see which tile it falls in?" For hats and Spectres, this doesn't matter, because of course the whole point is that all the tiles are the same. But for handling the 'legacy' aperiodic tilings – Penrose, Ammann-Beenker and others – it does matter!

Open ##2454892