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We systematically explore the structural and electronic properties of twistedtrilayer graphene systems. In general, these systems are characterized by twotwist angles, which lead to two incommensurate moir\'{e} periods. We show thatlattice relaxation results in the formation of domains of periodicsingle-moir\'{e} structures only for twist angles close to the simplestfractions. For the majority of other twist angles, the incommensurate moir\'{e}periods lead to a quasicrystalline structure. We identify experimentallyrelevant magic angles at which the electronic density of states is sharplypeaked and strongly correlated physics is most likely to be realized.

Multi-moir\'{e} trilayer graphene: lattice relaxation, electronic structure, and magic angles