The Plastic Number
$\rho\approx1.3247$ is the minimal PV number.
It is the unique real solution of the cubic equation
The three letter substitution rule whose scaling is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$. Though it might not look it at first glance, the Rauzy fractal is connected. This can be shown using the method of A. Siegel described in [Sie04]. The Rauzy fractal:
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles Plastic Number
Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Plastic Number
One possible version of a substitution rule with a free parameter: The upper tip of the three triangles can be shifted arbitrarily in horizontal direction, the result is always a self-similar substitution. The inflation multiplier is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.
Self Similar Substitution Polytopal Tiles Plastic Number