Geometry of the toroidal N-helix: optimal-packing and zero-twist

Two important geometrical properties of N-helix structures are influenced by bending. One is maximizing the volume fraction, which is called optimal-packing, and the other is having a vanishing strain-twist coupling, which is called zero-twist. Zero-twist helices rotate neither in one nor in the other direction under pull. The packing problem for tubular N-helices is extended to bent helices where the strands are coiled on toruses. We analyze the geometry of open circular helices and develop criteria for the strands to be in contact. The analysis is applied to a single, a double and a triple helix. General N-helices are discussed, as well as zero-twist helices for N > 1. The derived geometrical restrictions are gradually modified by changing the aspect ratio of the torus.