Iteration of rhombs: filling the gap (2)

First iteration of the rhomb.

The gaps in the fractal design of 12-fold rotational symmetry of my earlier post “Iteration of rhombs” can be filled in similarly as for the design of 8-fold rotational symmetry (see the post ” … filling the gap “.) Here we use two equilateral triangles instead of a square. Thus the rhomb is replaced by four smaller rhombs and two triangles. Then, there is no gap left.

First transformation of the triangle.

The next step then replaces each triangle by three rhombs and a three-pointed star. After these two steps the rhomb results in 15 rhombs and two stars without gap.

A rhomb after two iteration steps.

The triangle after two iterations.

In the next step, six smaller rhombs and four triangles replace the three-pointed star. The triangle is thus after two steps made of triangles and rhombs. With this method we get a tiling of 12-fold rotational symmetry containing rhombs, triangles and stars.

Quasiperiodic tiling of 12-fold rotational symmetry without squares.

There are surprising similarities with the earlier results for eight-fold rotational symmetry. Here, the length of the sides of the tiles changes each step by sqrt(2+sqrt(3)). The distance between the stars of 12 rhombs is (2+sqrt(3))*sqrt(2+sqrt(3)) = 7.14. The centers of these stars seem to be the corner points of the same tiling rotated by 15 degrees. Thus the tiling would be self-similar. It is quite different to the Stampfli tiling.

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