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. The 2D projection of all atomic positions in the β‐Mn unit cells shows that they are situated on three circumferences containing 2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. The exhaustive description of the non‐crystallographic symmetry of the β‐Mn crystal has been achieved by using the 8D E8 lattice in which both the 4D {3, 3, 5} polytope and cubic 6D B6 lattice can be inserted.