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Universal classification of twisted, strained and sheared graphene moiré superlattices

Moiré superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene's electronic properties. Yet, the exact crystallographic structure on which their band structure depends remains highly debated. In this scanning tunneling microscopy and density functional theory study, we have analysed graphene samples grown on multilayer graphene prepared onto SiC and on the close-packed surfaces of Re and Ir with ultra-high precision. We resolve small-angle twists and shears in graphene, and identify large unit cells comprising more than 1,000 carbon atoms and exhibiting non-trivial nanopatterns for moiré superlattices, which are commensurate to the graphene lattice. Finally, a general formalism applicable to any hexagonal moiré is presented to classify all reported structures.