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authorSimon Tatham <anakin@pobox.com>2023-03-31 18:35:43 +0100
committerSimon Tatham <anakin@pobox.com>2023-03-31 19:29:28 +0100
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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.
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