Spiral tetrahedral packing in the β-Mn crystal as symmetry realization of the 8D E₈ lattice

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.