Emergent quasiparticles in Euclidean tilings

A material's geometric structure is a fundamental part of its properties. The honeycomb geometry of graphene is responsible for its Dirac cone, while kagome and Lieb lattices host flat bands and pseudospin-1 Dirac dispersion. These features seem to be particular to a few 2D systems rather than a common occurrence. Given this correlation between structure and properties, exploring new geometries can lead to unexplored states and phenomena. Kepler is the pioneer of the mathematical tiling theory, describing ways of filling the Euclidean plane with geometric forms in his book Harmonices Mundi. In this article, we characterize 1255 lattices composed of k-uniform tiling of the Euclidean plane and unveil their intrinsic properties; this class of arranged tiles presents high-degeneracy points, exotic quasiparticles and flat bands as common features. Here, we present a guide for the experimental interpretation and prediction of new 2D systems.