Non-crystallographic helices in polymers and close-packed metallic crystals determined by the four-dimensional counterpart of the icosahedron

Abstract

Non-crystallographic fractional screw axes are inherent to the constructions of n-dimensional crystallography, where 3 < 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 three-dimensional Euclidean space {\bb E}^{3} are considered. These determine the structure of rods composed of deformed tetrahedra in close-packed crystals of α-Mn and β-Mn. In the {3, 3, 5} polytope, highlighted here for the first time, is a 40-vertex helix with a 20/9 axis composed of seven-vertex quadruples of tetrahedra (tetrablocks), whose 7/3 approximants determine in a crystal of an α-Mn 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 {\bb E}^{3}, 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 {\bb E}^{3} correspond to experimentally determined parameters of the α-helix, which allows us to explain the versatility of the α-helix in proteins by the symmetry of the polytope. The set of fractional axes of all periodic approximants of helices with 30/11, 20/9, 40/9, 40/11 axes, as well as the powers of these axes, are combined into a tetrahedral-polytope class of 50 basic axes. The basic axes as well as composite (defined as a combination of 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.