A long time ago I found a coloring of the rhombs of the Ammann-Beenker tiling using two colors such that translations exchange colors, see “two-fold color symmetry …“. In particular, there are stars of rhombs of both colors. They define the same coloring of the enlarged Ammann-Beenker tiling at the self-similarity ratio of 1+sqrt(2). Is there a similar coloring for the tiling of 10-fold rotational symmetry ?
This coloring can be used for all tilings, that have only tiles with an even number of sides. Each corner point of the tiling needs an extra variable. I gave it the name “sublattice” because it relates to a coloring of a hypercubic lattice. For some point of the tiling we simply put “sublattice=1”. The values for the other points are found by passing through the tiling. Going along an edge changes the sign of the value for “sublattice”. Thus all points that can be reached from the first point with an odd number of steps get the value -1. For an even number of steps we get +1. Note that this cannot be done if we have tiles or meshes with an odd number of sides.
The rhombs of the tiling get the first color if the corner points at their acute angles have the sublattice value +1. If the sublattice value is -1 we use the second color. Squares have only one color as their angles are all equal.
For the quasiperiodic tiling of 10-fold rotational symmetry I get this coloring:
I have used red and blue as the two basic colors. For the narrow rhombs I have mixed in a bit of black and for the fat rhombs a bit of white.
All stars of 10 rhombs are blue. Thus there is no color symmetry upon translation and the Ammann-Beenker tiling is probably a rather special case.
As in “tired of rhombs ?” we can get an alternative tiling, which is an approximate dual to this one. The corner points of the new tiling are at the centers of the rhombs. Edges connect points at centers of rhombs that have a common edge. Then each corner point of the first tiling has a tile of the new tiling and its sublattice variable determines the color of the tile. The tiling above gives this quasiperiodic checkerboard pattern: