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The chain of algebraic geometry constructions permits to transfer from theminimal surface with zero instability index, and from the lattice over the ringof cyclotomic integers to the tetra-block helix. The tetra-block is the7-vertex joining of four tetrahedra sharing common faces; it is considered as abuilding unit for structures approximated by the chains of regular tetrahedra.The minimality condition of the 7 - vertex tetrablock as a building unit is theconsequence of its unique mapping by the Klein's quartic (which ischaracterized by the minimal hyperbolic Schwartz triangle) into the minimalfinite projective geometry. The topological stability of this helix provided bythe pitch to radius ratio H/R of 2{\pi}/({\tau}+1) ({\tau} is the goldensection) and by the local rotation axis order of 40/11=40exp(-H/R). Theseparameters determine the helix of C{\alpha} atoms inside the alpha - helix withthe accuracy of up to 2%. They explain also the bonding relationship i -- i+4between the i-th amide group and the (i+4)-th carbonil group of the residues inthe peptide chain and the observed value of the average segment length of thealpha-helix which is equal to 11 residues. The tetra-block helix with the N,C{\alpha}, C', O, H atoms in the symmetrically selected positions, determinesthe structure of the alpha - helix. The proposed approach can displayadequately the symmetry of the helicoidal biopolymers.

Symmetrical-geometry constructions defining helicoidal biostructures. The case of alpha-helix