We consider self-similarities and iteration methods for quasiperiodic tilings of 12-fold symmetry. In “Iteration of rhombs: filling the gap (2)” the ratio of the lengths at each iteration step is sqrt(2+sqrt(3))=sqrt(3.73)=1.91. The Stampfli tiling has a self-similarity (see “Exhibition of the self-similarity of the tiling“) with a ratio of 7+4*sqrt(3) =(2+sqrt(3))*(2+sqrt(3))=13.8. Thus the irrational number 2+sqrt(3) seems to be characteristic for these tilings. This is similar to the number 1+sqrt(2) for tilings with eight-fold rotational symmetry.
To discover new tilings with twelve-fold rotational symmetry I wanted then to find an iterative scheme with the different ratio of 1+sqrt(3)=1.73 between the lengths at each step. This is a small number, which makes it rather difficult.
A rhomb is dissected into two smaller rhombs, one square and four halves of equilateral triangles. Note that the triangles will have different dissections.
In the following I have to consider quarters of the square, which too are squares. A quarter of a square is dissected into a rhomb, which has its acute corners at the center of the square and one of its corners, two quarters of a square and four halves of triangles.
For the halves of the triangles I need three different dissections. If there is a rhomb at its hypotenuse then it is dissected into a rhomb, one quarter of a square and four halves of triangles. The acute corners of the rhombs have to coincide.
If a square lies at the hypothenuse, then the dissection uses four halves of triangles and three quarters of a square.
These dissections fit nicely together to give a tiling of entire equilateral triangles and squares. It was quite difficult to have no lonely halves or quarters. Finally the resulting tiling is rather complex: