Iteration of rhombs

We had good weather the last days and I enjoyed myself in the garden – weeding and admiring the nice tulips. I also found a nice iterative method to draw a tiling, which I will share with you now. It is raining and I am back to blog again.

We simply use rhombs with an acute angle of 45 degrees. They can have two different colors, say red and blue. Looking at the Amman-Beenker tiling we find that a blue rhomb can be replaced by two smaller red rhombs and two blue rhombs, leaving a square hole at the center:

Similarly, a blue rhomb is replaced by the same design, but with exchanged colors:

Probably, a similar design is used by same company as its sign. Repeating this procedure on the above designs we get:

Doing some more iterations, we get some quite complex patterns. As an example I am showing you a magnified part of the tiling I got with eight iterations:

It has eight fold rotational symmetry with twofold color symmetry. It is quasiperiodic and self-similar because of the way it is made. If the iteration is further repeated, it will contain square-like holes of all sizes with a fractal boundary. Thus this tiling it is very different to the Amman-Beenker tiling.

Surprisingly, this is easy to program. Using processing I needed just 40 lines of code.

This entry was posted in Fractals, Tilings and tagged , , , , , , , . Bookmark the permalink.

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