Icosahedral Polyhedra from <em>D</em>₆ Lattice and Danzer’s <em>ABCK</em> Tiling

• Main Authors: Abeer Al-Siyabi, Nazife Ozdes Koca, Mehmet Koca
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Symmetry
• Subjects: lattices; Coxeter–Weyl groups; icosahedral group; projections of polytopes; polyhedra; aperiodic tilings
• Online Access: https://www.mdpi.com/2073-8994/12/12/1983

It is well known that the point group of the root lattice <em>D</em>₆ admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group <em>H</em>₃ , its roots, and weights are determined in terms of those of <em>D</em>₆ . Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of <em>D</em>₆ determined by a pair of integers (<em>m</em>₁,<em>m</em>₂) in most cases, either both even or both odd. Vertices of the Danzer’s <i>ABCK</i> tetrahedra are determined as the fundamental weights of <em>H</em>₃ , 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 (<em>m</em>₁,<em>m</em>₂) with coefficients from the Fibonacci sequence. Tiling procedure both for the <i>ABCK</i> tetrahedral and the <<em>ABCK</em>> octahedral tilings in 3D space with icosahedral symmetry <em>H</em>₃, and those related transformations in 6D space with <em>D</em>₆ symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in <em>D</em>₆. The tetrahedron <i>K</i> constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “<i>K</i>-polyhedron”, “<i>B</i>-polyhedron”, and “<i>C</i>-polyhedron” generated by the icosahedral group have been discussed.