0xDE (![[info]](http://l-stat.livejournal.com/img/userinfo.gif){kind=small} 11011110) wrote,@ 2006-09-11 19:33:00 |
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| Entry tags: | color, geometry, symmetry, tiling, unsolved |
Periodic coloring of tilings
The floret pentagonal tiling is shown below with two colorings. On the left is an optimal coloring: it uses the fewest colors possible, six, among colorings that respect all translational symmetries of the tiling. The quotient of the plane tiling by these symmetries is a tiling of the torus by six tiles, in which each tile touches each other, so six colors is optimal. But on the right, we have a periodic tiling with only three colors! The trick is that it has fewer symmetries: only 1/3 of the symmetries of the uncolored tiling preserve the colors in the coloring on the right.
What can be said about this kind of problem more generally? If one has a periodic tiling of the plane, how few colors are required to color it periodically, and what can be said about the index of the coloring's symmetry group as a subgroup of the uncolored tiling's symmetries? The four-color theorem together with a standard application of König's lemma shows that any tiling has a four-coloring, but I don't see how to force the coloring to be periodic and still use only four colors in general.
ETA: Five colors suffice. Still unclear whether four colors are always possible.
