The 3D facets of the Delone cells of the root lattice D₆ which tile the 6D Euclidean space in an alternating order are projected into 3D space. They are classified into six Mosseri–Sadoc tetrahedral tiles of edge lengths 1 and golden ratio τ = (1 + 5^1/2)/2 with faces normal to the fivefold and threefold 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 from six fundamental tiles, the faces of which are normal to the fivefold 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 τ generated by an inflation matrix. It is noted that dodecahedra with edge lengths of 1 and τ naturally occur already in the second and third order of the inflations. The 3D patches displaying fivefold, threefold and twofold symmetries are obtained in the inflated dodecahedral structures with edge lengths τⁿ with n ≥ 3. The planar tiling of the faces of the composite tiles follows the edge‐to‐edge matching of the Robinson triangles.

中文翻译：

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带有Mosseri–Sadoc瓷砖的十二面体结构

将根格子D ₆的Delone单元的3D面以交替的顺序平铺6D欧式空间的3D面投影到3D空间中。它们被分类为六个Mosseri–Sadoc四面体，其边长为1，黄金比率τ=（1 + 5 ^1/2）/ 2，面垂直于五倍轴和三倍轴。从六面体组的基本重量获得顶点的二十面体，十二面体和二十面体的二十面体按六个四面体进行剖析。一组四个图块由六个基本部分组成瓷砖，其面垂直于二十面体组的五倍轴。结果表明，复合3D欧几里德空间可以面对面平铺，从而最大程度地覆盖复合瓷砖，并由膨胀矩阵生成膨胀系数τ。注意，边缘长度为1和τ的十二面体自然已经出现在膨胀的第二和第三阶中。所述3D显示补丁五倍，在膨胀的十二面体结构具有边缘长度τ而获得三倍和两倍对称性^Ñ与Ñ ≥3.平面平铺瓦片复合的面的如下罗宾逊三角形的边缘到边缘的匹配。