A. A. Polyakov.
ABSTRACT
The 3D Penrose tiling (3DPT) is a quasiperiodic lattice with icosahedral symmetry, it is used in the crystallography of quasicrystals. The work aims to formulate the principles of selecting fractals for repeating 3DPT. The building elements of the fractals are the great (I) and the small (D) stellated dodecahedrons. An absolute inflationary approach to fractal construction is used. Each point of both 3DPT and the fractals is described by a 6-index. These six indices are vector components on a non-orthogonal basis, with 6 unit vectors directed from the centre to the vertices of the regular icosahedron. The fractal points can be classified into four types according to two criteria: integer or half-integer 6-indices, an even or odd sum of six indices. Multiplying the vectors of fractal points by the golden ratio leads to a cyclical change in these types. The rules for modification of these types when summing vectors are formulated. These rules made it possible to determine the types of points of each prefractal. For 3DPT repetition, prefractals with integer indices having an even and odd sum are chosen. It was proposed a method for determining the minimum 3D star of the fractal. Based on these principles, the points of the fractal 2I3I4I… are analysed. When choosing this fractal, no extra points that are not part of the 3DPT were found. This approach allowes 85% of the 3DPT points to be repeated.
KEYWORDS
fractals, three-dimensional Penrose tiling, quasiperiodic lattice, great stellated dodecahedron, small stellated dodecahedron
DOI:10.31857/S0235010622020062