  We study the lattice configuration and electronic structure of a double
moir\'{e} superlattice, which is composed of a graphene layer encapsulated by
two other layers in a way such that the two hexagonal moir\'{e} patterns are
arranged in a dodecagonal quasicrystalline configuration. We show that there
are between 0 and 4 such configurations depending on the lattice mismatch
between graphene and the encapsulating layer. We then reveal the resonant
interaction, which is distinct from the conventional 2-, 3-, 4-wave mixing of
moir\'{e} superlattices, that brings together and hybridizes twelve degenerate
Bloch states of monolayer graphene. These states do not fully satisfy the
dodecagonal quasicrystalline rotational symmetry due to the symmetry of the
wave vectors involved. Instead, their wave functions exhibit trigonal
quasicrystalline order, which lacks inversion symmetry, at the energies much
closer to the charge neutrality point of graphene.
