ref: 1af1204b9c33c9c03b2e0fe66c2f07d9729cbc72
parent: 52d801a06a804244292f4a872eeaf5e84a9f70b1
author: Simon Tatham <anakin@pobox.com>
date: Fri Mar 31 14:35:43 EDT 2023
hat-test: option to generate four-coloured hat tilings. This commit is purely frivolous even by Puzzles standards, in that it's totally unrelated to any actual puzzle. But I know at least one person has already used the 'hat-test' tool in this code base to generate a patch of hat tiling for decorative purposes, so it's useful in its own right. Also, now that I've worked out _how_ to do this, it's a shame not to keep the code. Of course, any tiling of the plane _can_ be four-coloured, just by the Four Colour Theorem. But for a tiling with structure it's nicer if the colouring is related to the structure in some way. And there's a reasonably nice explicit construction that does just that: the paper introducing the tiling observes that if each reflected hat is fused with a particular one of its neighbours, the resulting tiling is graph-theoretically equivalent to a tiling of the plane by hexagons. And _that_ tiling can be three-coloured, in a unique way up to colour choices. This induces a four-colouring of the hat tiling in which the reflected hats have a colour to themselves, and everything else is coloured the same as its corresponding hexagon in the three-colouring. Actually implementing this turns out not to be too difficult using my coordinate system. I hand-wrote tables giving a patch of colouring for each of the four kitemaps; then, whenever two kitemaps meet, you can determine how the colours map to each other by looking at the overlapping tiles. So I can have hat-test work out the colour of each tile as it goes. So hat-test now supports a '--fourcolour' option to apply this colouring to the output tiling.