The first pair of images on this page show a tiling fractal corresponding to the minimal Pisot. This is reverse engineered from an image presented by Shigeki Akiyama. Its construction is shown in more detail elsewhere.
In the construction of the attractor the larger element is rotated by a. If it is rotated by p-a instead, the following attractor is generated, which tiles the plane with a single copy in the unit cell.
In both the above constructions the smaller element is rotated by 5a. If, in the first construction this is rotated by 5a+p, the following attractor is generated. This modification, when applied to the second construction, does not create a tile.
In the above attractors the transformation corresponding to the larger transform has to be applied 5 times to produce an image which is the same size as the smaller transform. There are also tiles in which the transformation corresponding to the larger transform needs to be applied only 3 times to produce this effect. In this case the contraction of the larger element is the root of c2+c6=1. Solving this numerically gives c~=0.826031. In a similar fashion as for the minimal Pisot tile, a value of 73.632° is obtained for a, giving the following attractor. The smaller element is rotated by 3a.
Replacing a by p-a gives the following attractor.
Replacing the rotation 3a of the smaller element by 3a+p gives the following attractors.
Source: The first is reverse engineered from an image presented by Shigeki Akiyama. The remainder were independently invented, but I have since found other references to some of them.
© 2002 Stewart R. Hinsley