Mechanics and Geometry for Carbon Nanotube Networks and Biomembrane Nanotube Networks

The recent progress in the mechanics and geometry for carbon nanotube networks and biomembrane nanotube networks are briefly summarized. At equilibrium states, both the spontaneous Y-branched carbon nanotubes and the three-way biomembrane nanotube junctions are found to have identical symmetries. Two origins for these symmetries are revealed: One is the symmetrical differential geometry we proved, and another is the minimal energy principle. From the minimal energy principle, we reveal the stable equilibrium theorem for the networks formed by the spontaneous Y-branched carbon nanotubes or the three-way biomembrane nanotube junctions: A mechanically stable network is geometrically equivalent to a Steiner minimal tree with geometrical symmetries and topological invariabilities.