I have caught a cold. I am not able to do new work and thus I am writing up some old left-overs.
It is not possible to find an iteration method for the Stampfli tiling. One finds easily how to dissect a rhomb into squares, equilateral triangles and rhombs. But then there are many ways to dissect the squares and triangles because a triangle together with a square can be replaced by two rhombs and a displaced triangle. No simple rule exists to tell if we should take the triangle and square combination or rhombs and a triangle. But it is interesting to try the impossible if we thus discover something new.
We first try dissections of the squares and the triangles with a maximum of rhombs and a minimum of squares. This gives us a tiling with stars of twelve rhombs surrounded by sqaures, triangles and some rhombs making the same rosette as in the Stampfli tiling. But then there are far too many stars of rhombs and too few squares:
Alternatively, we can use another dissection of the triangle without rhombs. Together with the same dissection of the square we get a new tiling. But now, the tiling has too many squares and there are no stars of twelve rhombs. Again, this is not the Stampfli tiling: