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.
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.
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.
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.