The minimal Pisot tile is a plane-tiling figure which is composed of 2 copies of itself, of different sizes. The left hand image shows the division into 2 copies. The right hand image is a coloured rendition of a dissection published by Shigeki Akiyama.
It takes a little thought to work out an IFS which generates this figure.
The larger element can be recursively split into pairs of elements. This results in a spiral of elements, and a residual central element. When this has been done 4 times the central element is congruent to the original smaller element.
The transform for the larger element consists of a contraction and a rotation. Let us denote the transforms for the original two elements as T1 and T2. The transform for the central red element is T15, so we have
We can arbitrarily fix x as the vector (1,0).
T1 consists of contraction by c and rotation by a.
We know that c2+c10=1. Solving this numerically gives c~=0.868837.
Drawing lines at angles a and 3a through the points (0,0) and (1,0) gives us a triangle which shows that csina=c³sin(3a). Solving this numerically gives a~=220.328°.
The minimal Pisot tile tiles the plane. There are two copies in the minimal unit cell, with a ratio of areas of 1:c³.
A triangle whose sides have the ratios 1:c:c³ can be drawn connecting vertices of this tiling. The angles of this triangle can be obtained by the application of the Cosine Rule. The angle q, made by lines joining corresponding points of the red and blue copies is related to a by the equation q=3a mod 2p. The value of a derived above is equal to (2p+q)/3.
Source: Reverse engineered from an image published by Shigeki Akiyama.
References
© 2001, 2002 Stewart R. Hinsley