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Spiral tetrahedral packing in the β‐Mn crystal as symmetry realization of the 8D E 8 lattice
Author(s) -
Talis Alexander,
Everstov Ayal,
Kraposhin Valentin
Publication year - 2021
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273320012978
Subject(s) - icosahedral symmetry , tetrahedron , crystallography , crystal structure , lattice (music) , polytope , hexagonal lattice , combinatorics , crystal (programming language) , space group , geometry , physics , chemistry , materials science , mathematics , condensed matter physics , diffraction , x ray crystallography , quantum mechanics , antiferromagnetism , computer science , acoustics , programming language
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 E 3 . 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 E 3 space system of icosahedra from the 6D cubic B 6 lattice. The recently proposed determination of non‐crystallographic symmetry of the β‐Mn crystal actually uses the symmetries of the 8D E 8 lattice, in which both the 4D {3, 3, 5} polytope and cubic 6D B 6 lattice can be inserted.