## three-color symmetry from doubling the tiling of rhombs

I am going back to color-symmetries of quasiperiodic tilings. In “Three-color symmetry in rotation for the Stampfli tiling” I have shown that the twelve-fold rotational symmetry can exchange three colors if triangles are not considered.  A tiling of twelve-fold rotational symmetry containing only rhombs and squares could thus have perfect three-color symmetry. We find such a tiling by doubling the periodic tiling of rhombs with an acute angle of 60 degrees.

I am reusing the material of the post “Doubling the tessellation of hexagons and triangles“. There the tiling of rhombs generated the semiregular tessellation of hexagons and triangles. Now I go the other way. The tessellation of hexagons and triangles used as a grid makes the tiling of rhombs. Two such tessellations rotated by 90 degrees define a quasiperiodic tiling of rhombs and squares only. This tiling has lines across thin rhombs of 30 degrees acute angle and squares only. Thus I use red, green and blue for these tiles. The broader rhombs with an angle of 60 degrees lie between the thin rhombs. They too form lines together with the sides of tiles. Therefore I am using the secondary colors yellow, cyan and magenta for these rhombs:

Quasiperiodic tiling with three-color twelve-fold rotational symmetry.

Considering the stars of twelve rhombs, this tiling is not self-similar and not a decoration of the Stampfli tiling or the Socolar tiling.

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