Core/shell heterostructured conical carbon fibers with silicon-based cores and graphitic carbon sheaths have been grown on silicon carbide (SiC) surfaces exposed to the combustion of methane. The conical carbon fibers (CCFs), which taper from a circular base to a point, appear to form by two simultaneous growth processes, axial tip-growth and radial fiber thickening. Radial fiber thickening results from pyrolytic carbon deposition from gas-phase hydrocarbons. Oxidation of SiC surfaces is proposed to produce gas-phase silicon oxides, which feed axial tip-growth and install the silicon-based core. The unusual morphology of the CCFs, including bending, branching, curving and tapering, is proposed to result from the action and interplay of these two simultaneous processes.