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We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide.

We report in situ environmental transmission electron microscope observations of the nucleation and growth of multi-wall and single-wall carbon nanotubes formed by the catalytic decomposition of acetylene (C2H2) on Ni/SiO2 catalyst. The growth rate, structure and morphology of the carbon nanotubes formed depended upon reaction temperature and pressure. Under 20–100 mTorr of gas pressures at 480°C, serpentine-shaped or zigzag, multi-wall carbon nanotubes grew at an average rate of 35–40 nm sec−1. At pressures <10 mTorr at the same temperature, straight single-wall carbon nanotubes with nearly uniform diameters (∼3.5 nm) formed at average growth rates of 6–9 nm sec−1. The growth of both straight and serpentine carbon nanotubes tends to proceed at non-uniform rates, with frequent pauses followed by growth spurts. The nanotubes frequently contained sharp bends that turned the nanotube axis by ∼60° and ∼120°. We conjecture that the bends are related to the change in growth direction that is dictated by the crystallographic orientation of the catalyst particle. The rotations of the nanometer-sized catalyst particle may be caused by transient melting-recrystallization events caused by local thermal variations. The nanotube attempts to follow the preferred growth direction, while simultaneously attempting to maintain a seamless 3-coordinated graphene wall. This latter condition is most easily satisfied by the introduction of pentagon–heptagon defect pairs dissociated to the opposite sides of the nanotube creating the 60° bend. [ABSTRACT FROM AUTHOR]