The nucleation of curved graphitic structures such as carbon and boron nitride nanotubes in the absence of catalyst particles is still poorly understood. The nucleation and growth mechanisms of graphitic cones have been even more elusive. We investigate the formation of helical conical fibers of boron nitride (BN) and carbon (C) produced in a catalyst-free environment from the solid-state annealing of an amorphous BCN compound. Helical conical fibers consist of a single graphene sheet wrapped conically around an axis. We show that the cones grow radially in clusters, with each cluster arising from a common seed. The radial morphology originates from spherulite-like helicoidal polyhedral particles of BN that serve as templates for the BN and C cones. Evidence is presented that the original seed of the BN polyhedral particles is a ∼5 nm fullerene-like shell. We propose an idealized model for the seed structures that consists of multiply twinned hexagonal (h-)BN crystals containing a screw dislocation. This model provides additional insight into the view that conical graphitic structures originate from topological defects in graphene.