The assembly of uniform nanocrystal building blocks into well ordered superstructures is a fundamental strategy for the generation of meso- and macroscale metamaterials with emergent nanoscopic functionalities(1-)(10). The packing of spherical nanocrystals, which frequently adopt dense, face-centred-cubic or hexagonal-close-packed arrangements at thermodynamic equilibrium, has been much more widely studied than that of non-spherical, polyhedral nanocrystals, despite the fact that the latter have intriguing anisotropic properties resulting from the shapes of the building blocks(11-13). Here we report the packing of truncated tetrahedral quantum dot nanocrystals into three distinct superstructures-one-dimensional chiral tetrahelices, two-dimensional quasicrystal-approximant superlattices and three-dimensional cluster-based body-centred-cubic single supercrystals- by controlling the assembly conditions. Using techniques in real and reciprocal spaces, we successfully characterized the superstructures from their nanocrystal translational orderings down to the atomic-orientation alignments of individual quantum dots. Our packing models showed that formation of the nanocrystal superstructures is dominated by the selective facet-to-facet contact induced by the anisotropic patchiness of the tetrahedra. This study provides information about the packing of non-spherical nanocrystals into complex superstructures, and may enhance the potential of self-assembled nanocrystal metamaterials in practical applications.