In an earlier post “Iteration of rhombs” I got a fractal design of 8-fold rotational symmetry starting with a rhomb of acute angles of 45 degrees. An iterative step then replaces each rhomb by four rhombs, leaving a square gap between them. The gaps became complicated fractal shapes. But at the third iteration, the gap is a square with sides of twice the length of the sides of the rhombs:
Based on this, we can fill the gap if we also use squares in the iteration. A rhomb is then replaced by a square in addition to four smaller rhombs.
At the next step, the square is replaced by four rhombs along its sides and a four pointed star-like shape.
After this step the rhomb is now made of rhombs, squares and stars.
The next step replaces the star by eight rhombs and four squares. In all, the square is now replaced by eight squares and sixteen rhombs. There are no gaps left.
After this step, the rhomb is still made of rhombs, squares and stars and it has no gaps.
Thus we obtain a quasiperiodic tiling of eight-fold rotational symmetry.
The length of the tiles changes at each step by a factor of sqrt(2+sqrt(2)). Thus it is not related to the Ammann-Beenker tiling. The distance between the stars of rhombs is (2+sqrt(2))*sqrt(2+sqrt(2)) = 6.3 times the length of the sides of the tiles. The centers of the stars of rhombs themselves seem to make up the same tiling rotated by 22.5 degrees. Thus this tiling might be self-similar. It is quite distinct to the tiling using octagons I presented earlier.
The gap in the fractal of rhombs with 12-fold rotational symmetry can also be filled out as I will show in one of the next posts.