In this paper we derive the general equilibrium equations of a polymer chain with a noncircular cross section by the variation of the free energy functional. From the equilibrium equation of the elastic ribbon we derive analytically the equilibrium c
onformations both of the helical ribbons and the twisted ribbons. We find that the pitch angle of the helical ribbons depends on the ratio of the torsional rigidity to the bending one. For the twisted ribbons, the rotation rate depends on the spontaneous torsion, which is determined by the elastic properties of the polymers. Our results for helical and twisted ribbons strongly indicate that the formation of these structures is determined by their elastic properties.
