Best packing of identical helices

Abstract

In this paper we prove the unique existence of a ropelength-minimizing conformation of the theta-spun double helix in a mathematically rigorous way, and find the minimal ropelength Rop(*)(theta) = -8 pi/t where t is the unique solution in [-theta, theta] of the equation 2 - 2 cos(t+ theta)= t(2). Using this result, the pitch angles of the standard, triple and quadruple helices are around 39.3771 degrees, 42.8354 degrees and 43.8351 degrees, respectively, which are almost identical with the approximated pitch angles of the zero-twist structures previously known by Olsen and Bohr. We also find the ropelength of the standard N-helix.