  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.
