Crystal structures of alpha and beta modifications of Mn as packing of tetrahedral helices extracted from a four‐dimensional {3, 3, 5} polytope

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. 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.