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arXiv · 1211.3686

Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. I. Root lattice E8 and the closed sequence of algebraic polytopes

M. I. Samoylovich·A. L. Talis
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Abstract

In the framework of algebraic topology the closed sequence of 4-dimensional polyhedra(algebraic polytopes) was defined. These polytopes were determined by the second coordination sphere of 8-dimensional lattice E8. The ordered non-crystalline structure is determined by a chain of constructions of algebraic topology: an algebraic polytope, a homogeneous manifold in a 3-dimensional Euclidean space E3, locally-homogeneous manifold, locally minimal surface, one-parameter family of helicoids, bundle (cover) with a base of cell complexes, local-lattice packing of cell complexes into a substructure of E3, determined by helicoids. The formalism being developed allows one to surmount restrictions of classical crystallography and to single out a class of ordered non-crystalline structures, invariant with respect to structures determined by the lattice E8. The topological stability of such substructures is determined by their relatedness to Weierstrass' representation, as well as the condition that the instability index of the surface equals zero. Formation of such structures corresponds to lifting a configuration degeneracy, and the stability of a state - to existence of a point of bifurcation.

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M. I. Samoylovich, A. L. Talis. 2012-10-12. Symmetrical laws of structure of helicoidally-like biopolymers in the framework of algebraic topology. I. Root lattice E8 and the closed sequence of algebraic polytopes. https://arxiv.org/abs/1211.3686

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