Construction of 3D Penrose tiling from 3D stars via Minkowski addition

The three-dimensional Penrose tiling (3DPT) is a fundamental model for describing the atomic structure of icosahedral quasicrystals. In this work, we propose a new approach to generating the 3DPT based on fractals constructed through an infinite Minkowski summation of similar discrete sets. Our approach involves the union of three distinct fractals: 1) a fractal generated by the Minkowski summation of vertices from great stellated dodecahedra, 2) a fractal generated by the Minkowski summation of vertices from small stellated dodecahedra, and 3) 2D fractals based on the Minkowski summation of the vertices of rhombic sets having the structure of Fibonacci inflation tiling. The sizes of the initial discrete sets are selected taking into account the requirements of the cut-and-project method for constructing 3DPT. The initial sets are chosen to be scaled by a factor of τ² relative to the corresponding sets that form the projection of a 6D cube onto 3D space, where τ = (1 + √5)/2 ≈ 1.618 is the golden ratio. At each step of the summation process, the initial set is scaled by a factor of τ and then summed with the result from the previous Minkowski addition step. The properties of the points obtained as a result of Minkowski summation (such as integer or half-integer indices, and the parity of the sum of their 6D indices) are determined by the symmetry of the Klein four-group. For the construction of the 3DPT, only points with integer indices were used. When analyzing a 2D fractal, the central symmetry of both the 2D and 1D Fibonacci sequences is shown.