Twofold color symmetry in translation

I was a bit disappointed by results of the twofold color rotational symmetry shown in the previous post. Thus I was searching further and found something more interesting.

Amman-Beenker tiling with points of two colors from projection of a two-color hypercubic lattice.

In the earlier post “How to find these corner points of the tiles” we have seen that each point is defined by four integer numbers i1, j1, i2, j2. These numbers form a regular hypercubic lattice in four-dimensional space and the Amman-Beenker tiling is a projection of a part of this lattice. Similarly as for a two-dimensional checkerboard we can use two colors for the hypercubic lattice. For each point we calculate the sum of the integers S=i1+j1+i2+j2. Then, if a point has an even sum S it gets the first color, and if the sum is odd it gets the second color.

We can use the same choice of colors for the Amman-Beenker tiling. Each edge has points of both colors at its end and thus has no particular color. The rhombs have always points of the same color at the corner with an acute angle. Thus we can use this color for the rhomb, which then has two distinct colors. The squares have just one color as all angles are equal. The modified tiling has large patterns:

Amman-Beenker tiling with two color translational symmetry.

Looking at smaller patterns we find some distance away the same pattern with exchanged colors; red rhombs become yellow and yellow rhombs become red. Thus we have a twofold color symmetry in translation.

This tiling is self-similar as the basic Amman-Beenker tiling including the new colors ! Let us look at the yellow or red stars of eight rhombs. They form the same pattern as the yellow and red dots in the first drawing of this post. Thus we can get the same tiling with larger tiles using the stars as their edges:

Selfsimilarity of the tiling with twofold color symmetry in translation.

Here the red lines show the edges of the inflated tiling. Rhombs are either bleaching out or darkening the colors of the first tiling. This indicates the two colors of the rhombs. The squares, having just one color, leave the colors unchanged.

We see that light rhombs are always divided in the same way into yellow and red rhombs and squares. The color of the rhombs is just exchanged for the dark rhombs. The large squares have two different colorings of their subdivision into small tiles. Thus we could even use two different colors for the squares of the large tiling. This has yet to be examined.

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