I want to have two colors for each tile of the quasiperiodic tiling with 12-fold rotational symmetry presented in an earlier post. The colors of the tiles depends on their orientation and they should exchange if the tiling is rotated by 30 degrees. The color does not change upon a rotation of 60 degrees, because two exchanges (such as from color one to color two and then from color two to color one) leave the colors effectively unchanged. This works well for the rhombs and equilateral triangles but for the squares we get a problem. After a rotation by 90 degrees and three exchanges of colors a square of color one should have the color two. But it now has the same orientation as before and should thus have color one. This is only possible if the two colors are actually the same for the square. Thus we can only use one color for the square.
To make an example, I am choosing blue and green for the triangles. This puts the accent on the two different triangular grids arising from the two hexagon grids. We may think that the tiling is made of these triangles, which are cut apart. Rhombs and squares then fill up the gaps between the triangles. Thus I used different yellows for these tiles.
If you would like to see other colors you are welcome to make a comment with your proposal.