Experimental values of atomic positions in the β‐Mn crystal permit one to distinguish among them a fragment of the helix containing 15 interpenetrating distorted icosahedra, 90 vertices and 225 tetrahedra. This fragment corresponds to the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitive cubic lattice of these icosahedral helices envelopes not only all atoms of β‐Mn, but also all tetrahedra belonging to the tiling of the β‐Mn structure. The 2D projection of all atomic positions in the β‐Mn unit cells shows that they are situated (by neglecting small differences) on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. Non‐crystallographic symmetry of the β‐Mn crystal is defined by mapping the closed icosahedral helix of the {3, 3, 5} polytope into 3D Euclidean space E³. This interpretation must be correlated also with the known previous determination of non‐crystallographic symmetry of the β‐Mn crystal by mapping into the 3D E³ space system of icosahedra from the 6D cubic B₆ lattice. The recently proposed determination of non‐crystallographic symmetry of the β‐Mn crystal actually uses the symmetries of the 8D E₈ lattice, in which both the 4D {3, 3, 5} polytope and cubic 6D B₆ lattice can be inserted.

中文翻译：

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β-Mn晶体中的螺旋四面体堆积作为8D E8晶格的对称实现

β-Mn晶体中原子位置的实验值可让人们区分出包含15个互穿的扭曲二十面体，90个顶点和225个四面体的螺旋碎片。该片段对应于4D {3，3，5}多表位中15二十面体的闭合螺旋。这些二十面体螺旋的原始立方晶格不仅包裹着β-Mn的所有原子，而且包裹了所有属于β-Mn结构的四面体。β-Mn晶胞中所有原子位置的2D投影显示（忽略微小差异）它们位于三个圆周上，该圆周包含{3，3，5}多表位的90个顶点的2D投影在同一平面上。通过将{3，3，5}多表位的闭合二十面体螺旋映射到3D欧几里得空间中来定义β-Mn晶体的非晶体对称性E ³。这种解释还必须与先前已知的β-Mn晶体非晶体对称性测定方法相关，方法是从6D立方B ₆晶格映射到二十面体的3D E ³空间系统中。最近提出的确定β-Mn晶体非晶体对称性的方法实际上是使用8D E ₈晶格的对称性，其中可以插入4D {3，3，5}多晶格和立方6D B ₆晶格。