On the perfect hexagonal packing of rods

In most cases the hexagonal packing of fibrous structures or rods extremizes the energy of interaction between strands. If the strands are not straight, then it is still possible to form a perfect hexatic bundle. Conditions under which the perfect hexagonal packing of curved tubular structures may exist are formulated. Particular attention is given to closed or cycled arrangements of the rods like in the DNA toroids and spools. The closure or return constraints of the bundle result in an allowable group of automorphisms of the cross-sectional hexagonal lattice. The structure of this group is explored. Examples of open helical-like and closed toroidal-like bundles are presented. An expression for the elastic energy of a perfectly packed bundle of thin elastic rods is derived. The energy accounts for both the bending and torsional stiffnesses of the rods. It is shown that equilibria of the bundle correspond to solutions of a variational problem formulated for the curve representing the axis of the bundle. The functional involves a function of the squared curvature under the constraints on the total torsion and the length. The Euler-Lagrange equations are obtained in terms of curvature and torsion and due to the existence of the first integrals the problem is reduced to the quadrature. The three-dimensional shape of the bundle may be readily reconstructed by integration of the Ilyukhin-type equations in special cylindrical coordinates. The results are of universal nature and are applicable to various fibrous structures, in particular, to intramolecular liquid crystals formed by DNA condensed in toroids or packed inside the viral capsids