A special three-color symmetry

Earlier I have presented a coloring of the Amman-Beenker tiling which exchanges colors upon translation, see “Twofold color symmetry in translation” and “Twofold color symmetry in translation – revisited“. It resulted from the square checkerboard. As there are many similarities between the Amman-Beenker and the Stampfli tilings I wanted to create something similar for the Stampfli tiling using the projection method.

The Stampfli tilling results from the superposition of two hexagon grids as discussed in “A tiling of 12-fold rotational symmetry from two hexagon grids“. For hexagon grids there is a kind of checkerboard pattern with three colors. Each hexagon is surrounded by hexagons with the other two colors. Their colors alternate.

Coloring the hexagon grid with three colors similarly as a square checkerboard.

Now I need the mathematical details presented in “How to calculate the corner points and the projectionÂ method” to get the color of the hexagons. The center of a hexagon is determined by indices i and j. We simply add these indices and get the remainder of the sum by a division by three. The result is either 0,1 or 2 and determines the color for the center point and the hexagon. I am showing an example using yellow, orange and red. Note that a square checkerboard can be made quite similarly.

Now we combine two hexagon grids. The corner points of the resulting tiling has the four numbers i1,j1,i2 and j2. Again we sum this number and calculate the remainder of a division by three. This gives the color of the corner points:

Corner points of the Stampfli tiling with the three different colors resulting from their indices.

Now each equilateral triangle has one corner point of each color. Thus we cannot choose between one of these colors and thus use a fourth color for all triangles. Then the squares and rhombs have two corner points of the same color and we use this color for the tile. This gives us the colored tiling:

Stampfli tiling with the special three color symmetry.

Surprisingly, this coloring seems to be self-similar. Just look at the stars of twelve rhombs. As discussed before, they are corner points of the same tiling at an inflated scale. There are always six rhombs of the same color in such a star. If we use this color for the corner point at the inflated scale we also get the same coloring of the inflated tiling as shown here.

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