Finding an iteration method for the Stampfli tiling – mission impossible

I have caught a cold. I am not able to do new work and thus I am writing up some old left-overs.

Dissection of a rhomb.

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.

Dissection of a triangle with a maximum of rhombs.

Dissection of a square using a maximum of rhombs.

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:

Quasiperiodic tiling of rhombs, triangles and squares with a maximum of rhombs.

Dissection of a triangle without rhombs.

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:

Quasiperiodic tiling with many squares and few rhombs.

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