  Since the first-principles calculations in quantum chemistry precisely
provide possible configurations of carbon atoms in nanocarbons, we have
analyzed the geometrical structure of the possible carbon configurations and
found that there exists a novel symmetry in the nanocarbons, i.e., the
pre-constant discrete principal curvature (pCDPC) structure. In terms of the
discrete principal curvature based on the discrete geometry for trivalent
oriented graphs developed by Kotani, Naito, and Omori (Comput. Aided Geom.
Design, $\bf{58}$, (2017), 24-54), we numerically investigated discrete
principal curvature distribution of the nanocarbons, C$_{60}$, carbon
nanotubes, C$_{120}$ (C$_{60}$ dimer), and C$_{60}$-polymers (peanut-shaped
fullerene polymers). While the C$_{60}$ and nanotubes have the constant
discrete principal curvature (CDPC) as we expected, it is interesting to note
that the C$_{60}$-polymers and C$_{60}$ dimer also have the almost constant
discrete principal curvature, i.e., pCDPC, which is surprising. A nontrivial
pCDPC structure with revolutionary symmetry is available due to discreteness,
though it has been overlooked in geometry. In discrete geometry, there appears
a center axisoid which is the discrete analogue of the center axis in the
continuum differential geometry but has three-dimensional structure rather than
a one-dimensional curve due to its discrete nature. We demonstrated that such
pCDPC structure exists in nature, namely in the C$_{60}$-polymers. Furthermore,
since we found that there is a positive correlation between the degree of the
CDPC structure and stability of the configurations for certain class of the
C$_{60}$-polymers, we also revealed the origin of the pCDPC structure from an
aspect of materials science.
