Toric geometry studies a simple class of algebraic varieties encoded by convex integral polytopes.
For example, the standard simplex corresponds to the projective space. I will describe a generalization of toric geometry which gives combinatorial counterparts of more complicated algebraic varieties, like curves, K3 surfaces etc. Natural examples come from Mirror Symmetry. The subject is related to non-archimedean geometry and motivic integration. The global regularity is proven using techniques of the Littlewood-Paley dyadic decomposition.