Symmetry of helicoidal biopolymers in the frameworks of algebraic geometry: α-helix and DNA structures

The chain of algebraic geometry and topology constructions permits one to single out a special class of discrete helicoidal structures. The symmetry of these structures determines the structural parameters of the α-helix and topology of the A, B and Z forms of DNA.

Finite noncrystallographic groups, 11-vertex equi-edged triangulated clusters and polymorphic transformations in metals

The one-to-one correspondence between a set of cosets of the Mathieu group M₁₁, block sets of the Steiner system S(4, 5, 11) and 11-vertex equi-edged triangulated clusters determines symmetry-allowable transformations between different 11-vertex triangulated clusters as automorphisms of the Steiner system S(4, 5, 11). The transformations can describe local reconstructions corresponding to polymorphic transformations in metals.