  We investigate generation of new Dirac cones in graphene under
double-periodic and quasiperiodic superlattice potentials. We first show that
double-periodic potentials generate the Dirac cones sporadically, following the
Diophantine equation, in spite of the fact that double-periodic potentials are
also periodic ones, for which previous studies predict consecutive appearance
of the cones. The sporadic appearance is due to the fact that the dispersion
relation of graphene is linear only up to an energy cutoff. We then show that
quasiperiodic potentials generate the new Dirac cones densely with its density
depending on the energy. We also extend the above predictions to other
materials of Dirac electrons with different energy cutoffs of the linear
dispersion.
