Boerdijk–Coxeter helix and biological helices as quasicrystals

Helices and dense packing of spherical objects are two closely related problems. The Boerdijk–Coxeter helix (B–C), a linear stacking of regular tetrahedra, is a very efficient solution to some close-packing problems, including protein folding. The structure of biological helices (α-helix, collagen) is determined chiefly by steric repulsion. Thus, maximum density leads directly to structures related to the B–C helix. We show that the B–C helix is a quasicrystalline structure with 1+√3 edges per turn, which can be disclinated and inflated. The collagen molecule consists of three intertwined helicoidal chains on the fibres of an inflated Hopf fibration. The central core is a B–C helical packing of the hydrogen atoms side-groups of the amino acid Gly.