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Using scanning-tunneling-microscopy and theoretical modeling onheterostructures of twisted bilayer graphene and hexagonal Boron-Nitride, weshow that the emergent super-moire structures display a rich landscape ofmoire-crystals and quasicrystals. We reveal a phase-diagram comprised ofcommensurate moire-crystals embedded in swaths of moire quasicrystals. The 1:1commensurate crystal, expected to be a Chern insulator, should only exist atone point on the phase-diagram, implying that it ought to be practicallyundetectable. Surprisingly we find that the commensurate crystals exist over amuch wider than predicted range, providing evidence of an unexpectedself-alignment mechanism that is explained using an elastic-network model. Theremainder of the phase-diagram, where we observe tunable quasicrystals, affordsa new platform for exploring the unique electronic-properties of these rarelyfound in nature structures.

Imaging Self-aligned Moir\'e Crystals and Quasicrystals in Magic-angle Bilayer Graphene on hBN Heterostructures