2020/03/31 by Abeer Al-Siyabi, Nazife Ozdes Koca, Mehmet Koca
Computer Science · Materials Science · Mathematics · #Cellular Automata and Applications #Fibonacci number #Group (periodic table) #Icosahedral symmetry #Lattice (music) #Mathematical Approximation and Integration #Penrose tiling #Polyhedron #Quasicrystal #Quasicrystal Structures and Properties #Symmetry group #Tetrahedron #math.MG #msc:20 #msc:52B10 #msc:52B11 #msc:52B20 #msc:52C07 #msc:52C22 #msc:52C23
paper · pdf · doi:10.3390/sym12121983
published as Symmetry 2020, 12, 1983 · 18 pages, 3 Tables, 12 figures
openalex created_date 2020/04/03 · openalex publication_date 2020/11/30 · arxiv created 2020/12/11 · arxiv updated 2020/12/14 · openalex updated_date 2026/08/05
It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.