Аннотация:Experimental values of atomic positions in the -Mn crystal permit one todistinguish among them a fragment of the helix containing 15 interpenetratingdistorted icosahedra, 90 vertices and 225 tetrahedra. This fragment correspondsto the closed helix of 15 icosahedra in the 4D {3, 3, 5} polytope. The primitivecubic lattice of these icosahedral helices envelopes not only all atoms of -Mn,but also all tetrahedra belonging to the tiling of the -Mn structure. The 2Dprojection of all atomic positions in the -Mn unit cells shows that they aresituated (by neglecting small differences) on three circumferences containing2D projections of 90 vertices of the {3, 3, 5} polytope on the same plane. Noncrystallographicsymmetry of the -Mn crystal is defined by mapping the closedicosahedral helix of the {3, 3, 5} polytope into 3D Euclidean space E3. Thisinterpretation must be correlated also with the known previous determination ofnon-crystallographic symmetry of the -Mn crystal by mapping into the 3D E3space system of icosahedra from the 6D cubic B6 lattice. The recently proposeddetermination of non-crystallographic symmetry of the -Mn crystal actuallyuses the symmetries of the 8D E8 lattice, in which both the 4D {3, 3, 5} polytopeand cubic 6D B6 lattice can be inserted.
