Post #2400401

It's not Tuesday any more, but here are a few more bonus pretty pictures. There's a clever construction called the "Wieringa roof" in which you assign every vertex of a P3 rhomb tiling a z-coordinate in {1,2,3,4}, with each rhomb having coordinates either 1,2,3,2 or 4,3,2,3, so that the tiling is lifted out of the flat plane into a crinkly surface. With the right scale factor this turns the two rhomb types into the _same_ shape of 3D rhombus, just tilted at different angles to the horizontal. So another way you can classify tiles in a P3 tiling is by their Wieringa height: do they have a vertex at the minimum height 1, or one at the maximum height 4? (Every tile has exactly one of these.) If you distinguish P3 tiles as low vs high _and_ by their 10 orientations, then it's not only possible to four-colour the adjacency graph, but to do it in such a way that every tile is a different colour from the other tile type at the same orientation (counting "same orientation" as the vector toward the extreme vertex from the opposite one), _and_ from the 180° rotations of both. This gives an almost balanced colouring (the density of the colour classes only vary by about 10%) and I think it looks much nicer than the previous version. You can assign vertex heights in the same kind of way to a P2 tiling, by considering its half-deflation to P3. That doesn't give you a nice 3D structure (the tiles would end up non-planar), but it still lets you classify kites and darts into two types each. With that and orientation, the adjacency graph still isn't four-colourable, but we can refine by neighbourhood as before, and again end up with a colouring that's better balanced and has fewer obvious regularities (though still a few). Here are the same two pictures as in the previous post, recoloured using these slightly nicer systems. As a bonus, I've also rendered the same two tilings by colouring _only_ by Wieringa height, so that you can see the rather nice patterns made by the high tiles and the low tiles.