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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.