  The recently realized bilayer graphene system with a twist angle of
$30^\circ$ offers a new type of quasicrystal which unites the dodecagonal
quasicrystalline nature and graphene's relativistic properties. Here, we
introduce a concise theoretical framework that fully respects both the
dodecagonal rotational symmetry and the massless Dirac nature, to describe the
electronic states of the system. We find that the electronic spectrum consists
of resonant states labeled by 12-fold quantized angular momentum, together with
the extended relativistic states. The resulting quasi-band structure is
composed of the nearly flat bands with spiky peaks in the density of states,
where the wave functions exhibit characteristic patterns which fit to the
fractal inflations of the quasicrystal tiling. We also demonstrate that the
12-fold resonant states appear as spatially-localized states in a finite-size
geometry, which is another hallmark of quasicrystal. The theoretical method
introduced here is applicable to a broad class of "extrinsic quasicrystals"
composed of a pair of two-dimensional crystals overlaid on top of the other
with incommensurate configurations.
