Two-dimensional packings of objects or corresponding polygonal tilings of the plane arise in a great variety of systems: biological tissues, foams, emulsions, polycrystals, granular materials, and many others. The structure of these systems is disordered, but far from arbitrary. We investigate which features of disorder can be used to diagnose differences between systems (e.g. pathological changes in tissues), to infer physical properties (e.g. interfacial tension between domains), or to correlate statistical quantities with morphological traits (e.g. anisotropy of cells). Using simple experiments and fundamental modeling ideas, we have successfully explained strong correlations between the distributions of domain sizes (areas) and topologies (number of neighbors), encompassing many living and inanimate systems. One consequence of these results is a direct explanation for the phenomenon of terminal polydispersity encountered in disordered close-packings in
2D: why is a certain amount of polydispersity necessary to obtain such packings? Another result reveals the first direct theoretical explanation for Lewis' law, a correlation between size and topology empirically known since the 1920s. We show that local domain shape and the statistics of the local domain environment are sufficient to explain these and other phenomena.