Twofold color symmetry in translation – revisited

In my earlier post “twofold color symmetry in translation” I used that the projection method defines four indices for each corner point of the tiles. The sum of the indices is either an odd or even number. Accordingly, the points belong to two different sets and have one of two colors. Then, the color of the corner point with an acute angle is used for the rhombs.

But this is too complicated and we have a much simpler approach. We can assign the number +1 or -1 to the corner points if we need that the two points connected by an edge of a tile should have opposite sign. This gives the same two sets of points as before. I got this idea by thinking of the tiling as a network with the corner points as nodes.

Subdivision of a yellow rhomb.

Subdision of an orange rhomb.

This can easily be used in the iterative method for making the Ammann-Beenker tiling. Note that this method is also discussed in Wikipedia (however I did not really understand the presentation). Here it is important that we do not use the longer side (the hypotenuse) of the triangles for assigning the numbers +1 or -1, as this line connects two points with the same sign. It is also nice that the subdivisions replace one side of the tiles by three sides of smaller tiles. Thus the iteration does not change the sign of already existing points.

Subdivision of a blue triangle.

Subdivision of a brown triangle.

The color of the rhombs is given by the points at their acute angles as before. But now we choose the color of the triangles from the sign of the point at its right angle. Thus we get the color scheme of the iterative method. As expected, the triangles that form together a square now have the same color.

Amman-Beenker tiling with twofold color symmetry in translation as obtained from the iterative method.

 

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