Affine extension of D_6 and Icosahedral Symmetric Quasicrystallography

Mathematical structures of the icosahedral quasicrystallography will be studied by projecting the Voronoi tessellation of the root lattice of D_6 defined by the affine extension of Coxeter group of D_6. First, we will construct the affine icosahedral subgroup H_3 of the affine Coxeter group D_6. This work will be an extension of the work done for affine H_2/A_4 system which describes the five-fold symmetric planar quasicrystallography. The Voronoi cells of the subgroups chain D_4? D_5 ? D_6 project as polyhedral substructures described by in the respective order as Bilinski dodecahedron ? rhombic icosahedron? rhombic triacontahedron. Projections of the facets of the Voronoi cells of the groups chain D_4? D_5 ? D_6 follows the chain structure as the golden rhombus?golden rhombohedra of two types ? Bilinski dodecahedron, all constitute the structures of icosahedral quasicrystallography. The group theoretical analysis will be developed and applied for the tessellation of the 3D space with rhombic icosahedron, rhombic triacontahedron, Bilinski dodecahedron and the acute golden rhombohedra.