具有Mosseri-Sadoc瓦的十二面体结构
Dodecahedral Structures with Mosseri-Sadoc Tiles
摘要：
将表示以交替顺序平铺六维欧几里得空间的根晶格D 6的深孔和浅孔的Delone单元的3D刻面投影到三维空间中。它们被分为六个边长为1和黄金分割比（tau）的Mosseri-Sadoc四面体瓦片，面垂直于5倍和3倍轴。以六个四面体为单位，对顶点由二十面体群的基本权求出的二十面体、十二面体和二十半十二面体进行了剖分。一组四个瓦片由六个基本瓦片组成，这些瓦片的面垂直于二十面体群的5重轴。结果表明，3D-欧几里得空间可以平铺面对面的最大面覆盖的复合瓷砖与膨胀因子τ产生的膨胀矩阵。我们注意到边长为1和τ的十二面体自然已经出现在暴胀的第二和第三阶中。在边长为n次方的膨胀十二面体结构中获得了显示5重、3重和2重对称性的3D面片，其中n等于3或大于3。复合瓦片的面的平面平铺遵循Robinson三角形的边到边匹配。
英文摘要：
3D-facets of the Delone cells representing the deep and shallow holes of the root lattice D6 which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio (tau) with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor tau generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and tau naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths tau to the power n with n equals 3 or greater than 3. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.
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