  We investigate the classical equilibrium properties and metamorphosis of the
ground state of interacting dipoles with fixed locations on a helix. The
dipoles are shown to align themselves along separate intertwined dipole chains
forming single, double, and higher-order helical chains. The number of dipole
chains, and their properties such as chirality and length scale on which the
chains wind around each other, can be tuned by the geometrical parameters. We
demonstrate that all possible configurations form a self-similar bifurcation
diagram which can be linked to the Stern-Brocot tree and the underlying Farey
sequence. We describe the mechanism responsible for this behavior and
subsequently discuss corresponding implications and possible applications.
