There is one semiregular tessellation of six-fold rotational symmetry left over which I have not yet used to create a quasiperiodic tiling of 12-fold rotational symmetry. It has rings of triangles such that the hexagons do not touch each other, see the figure at the left. Note that there is no mirror symmetry.
We then proceed as before and take two grids, rotated by 90 degrees with respect to each other. Then we let the computer build the dual of the combined grid. This gives us a quasiperiodic tiling of 12-fold rotational symmetry. It took me a lot of time to find a part of tiling that shows a nearly perfect symmetric part:
There is no classical exact 12-fold rotational symmetry. But you can take any finite part of the tiling and you will find exactly the same image rotated by 30 degrees somewhere else in the tiling. But this tiling is not mirror symmetric. If you take a sufficiently large part you will not find its mirror image anywhere in the tiling. Note the pairs of rhombs that seem to swim counter-clockwise around a common center.