The crystal structures of both α‐ and β‐Mn modifications have been presented as packing of tetrahedral helices extracted from four‐dimensional {3, 3, 5} polytope construction. Presentation of the β‐Mn structure as a primitive cubic arrangement formed by double tetrahedral helices around a central tetrahedral Coxeter–Boerdijk helix (tetrahelix) enables the inclusion in the structure description not only all atoms but also all tetrahedra; these tetrahedra are not accounted for in the preceding models for the β‐Mn structure. The tetrahelix periodicity arising by minimal deformations of tetrahedra edges is equal to eight tetrahedra and coinciding with the lattice periods of both modifications. The linear substructure of α‐Mn crystal consists of four tetrahelices which join to each other by edges around the common twofold axis. The α‐Mn structure has been presented as primitive cubic arrangement constructed from such rods.

中文翻译：

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Mn的α和β修饰的晶体结构，是从{3，3，5}四维多面体中提取的四面体螺旋的堆积

α-和β-Mn修饰的晶体结构已被展示为从三维{3，3，5}多表位构造中提取的四面体螺旋的堆积。将β-Mn结构表示为由围绕中心四面体Coxeter-Boerdijk螺旋（tetrahelix）的双四面体螺旋形成的原始立方排列的结构，不仅可以将所有原子都包括在内，而且还可以将所有四面体都包括在内。这些四面体在前面的β-Mn结构模型中未作说明。由四面体边缘的最小变形产生的四面体周期性等于八个四面体，并且与两种修饰的晶格周期一致。α-Mn晶体的线性子结构由四个四螺旋组成，它们通过围绕共同的双倍轴的边相互连接。