Intrinsically multilayer moiré heterostructures

We introduce trilayer and multilayer moir\'e heterostructures that cannot be viewed from the ``moir\'e-of-moir\'e'' perspective of helically twisted trilayer graphene. These ``intrinsically trilayer'' moir\'e systems feature periodic modulation of a local quasicrystalline structure. They open the door to realizing moir\'e heterostructures with vastly more material constituents because they do not constrain the lattice constants of the layers. In this manuscript, we define intrinsically multilayer patterns, provide a recipe for their construction, derive their local configuration space, and connect the visual patterns to physical observables in material systems.