Аннотация:Non-crystallographic fractional axes are inherent to the constructions of n-dimensional crystallography, three < n 8. This fact allows one to consider experimentally obtained helices as periodic approximants of helices from the four-dimensional {3, 3, 5} polytope and its derivative constructions.For the tetrahedral Coxeter–Boerdijk helix (tetrahelix) with a 30/11 axis from the {3, 3, 5} polytope, approximants with 11/4 and 8/3 axes in threedimensional Euclidean space E3 are considered. They[?] determine the structure of rods composed of deformed tetrahedra in close-packed crystals of -Mn and -Mn. In the {3, 3, 5} polytope we highlight for the first time a 40-vertex helix with an 20/9 axis composed of seven-vertex quadruples of tetrahedra (tetrablocks), whose 7/3 approximants determine in the crystal of -Mn a rod of deformed tetrablocks with the same period as the 11/4 approximant of the tetrahelix. In the spaces of the three-dimensional sphere and E3, the parameters of 20/9, 40/9 and 40/11 helices, as well as of their 20- and 40-vertex approximants, are calculated. The parameters of the approximant of the 40/11 helix in E3 correspond to experimentally determined parameters of the -helix, which allows us to explain the versatility of the -helix in proteins by the symmetry of the polytope. The set of fractional axes of all periodic approximantsof helices with 30/11, 20/9, 40/9, 40/11 axes, as well as the powers[multiples?] of these axes, are united into a tetrahedral-polytope class of 50 basis[basic] axes.The basis[basic] axes as well as composite (defined as unions[?better word needed? combination?] of basis[basic] ones) fractional axes of this class cover all fractional axes known to us according to literature data for polymers,biopolymers and close-packed metals. If the representation of a crystal by a lattice of helices with non-crystallographic fractional axes makes sense in terms of physics, then determination of its structural type requires an indication of its space group as well as the axis of the helix from the tetrahedral-polytope class.
