论文标题
来自D6晶格和Danzer的ABCK瓷砖的二十面体Polyhedra
Icosahedral Polyhedra from D6 lattice and Danzer's ABCK tiling
论文作者
论文摘要
众所周知，根晶格D_6的点组将二十面体组作为最大亚组。二十面体组H_3的发电机，其根和权重是根据D_6的生成器确定的。在大多数情况下，通过一对整数（M1，M2）确定的D_6晶格向量的投影获得了具有二十面体对称性的柏拉图式和阿基赛马固体。 Danzer的ABCK四面体的顶点确定为H3的基本权重，并且表明瓷砖的膨胀可以作为晶格矢量的投影获得，这些晶格向量的投影是由整数组合（M1，M2）具有fibonacci sequeffers sequence cefficients的线性组合（M1，M2）的特征。通过确定3D中的旋转和翻译以及d_6中的相应组元素，指定了H_3中ABCK四面体瓷砖和<ABCK>八面体瓷砖的平铺程序。四面体K构成二十面体群的基本区域，并在组作用时产生菱形三角体。已经讨论了由二十面体产生的K-Polyhedron，B-Polyhedron和C-Polyhedron的性质。
It is well known that the point group of the root lattice D_6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H_3, its roots and weights are determined in terms of those of D_6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D_6 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 Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral tiling and the <ABCK> octahedral tiling in H_3 and the corresponding D_6 spaces are specified by determining the rotations and translation in 3D and the corresponding group elements in D_6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of the K-polyhedron, B-polyhedron and the C-polyhedron generated by the icosahedral group have been discussed.
