Post #2454878

2026-05-08 15:46 UTC

@csk@mathstodon.xyz @pieter@mathstodon.xyz I certainly agree with your second paragraph – with bignums and a finite-degree field extension you can do all of this in exact arithmetic. "And I don't think you even have to go very far from the origin to see that" – that's a shame! I was hoping that the hat tiling being based on kites would help there, because you can separately calculate where you are in a kite and where that kite is in the tiling, so the quality of the FP approximation might degrade only per kite and not per pixel. I'd expect it to be harder in a Spectre version of this idea, where there's no discrete underlying grid. What's interesting to _me_, naturally, is how the two systems of identifying a tiling match up to each other. I represent a specific tiling via a sequence of supertile types and the relationship between each and its subtile; this technique represents one via a point in the plane. Translating between the two must be possible, but is probably also a royal pain. And I have to guess it won't be quite possible in some of the (alas, most interesting) edge cases, like the singular patterns. If it's anything like the pentagrid system for Penrose tilings, there won't be any real (x,y) value that _exactly_ represents a specific singular instance of the hat tiling. Instead, there will be an (x,y) value that represents the symmetric parts in common between more than one of them, and a full tiling that breaks the symmetry in a specific way can only be realised as the limiting tiling you obtain by approaching the singular point from a particular direction.

Replies (2)

@simontatham@hachyderm.io 2026-05-08 15:55

@csk@mathstodon.xyz @pieter@mathstodon.xyz … though thinking about it, that idea of "per kite and not per pixel" is nonsense, because in fact one surely rounds to the nearest point of the integer lattice that samples the fractal, which is neither of those. Posted before thinking.

Open ##2454879
@pieter@mathstodon.xyz 2026-05-08 19:35

@simontatham@hachyderm.io @csk@mathstodon.xyz There's definitely more work needed for tilings by spectres, or versions of the Hat family other than the hat or turtle, but since the hats-in-turtles tiling is based on a discrete grid, and because there exists a transformation of a tiling by spectres to a hats-in-turtles tiling that translates tiles by a finite distance, given a point, you just need to calculate a finite patch of hats-in-turtles which is guaranteed to cover the point once the patch is transformed to spectres. The relation between representations by points and sequences of supertiles is fairly simple if you ignore the edge cases. Any point inside the fractal region determines a nested sequence of fractal regions, each of which contains the point, and is one of the elements of the partition of the previous region, as shown in the diagram. So the list of supertile types and relations can be used to write the coordinates of the point as a \(\beta\) expansion of the form \(\sum_{n=0}^\infty c_n \phi^{-n}\) where (\phi\) is the square of the golden mean, and the \(c_n\)'s come from a finite set of complex numbers

Open ##2454882