E² and Gamma Distributions in Polygonal Networks

From solar supergranulation to salt flats in Bolivia, from veins on leaves to cells on Drosophila wing disks, polygon-based networks exhibit great complexities, yet similarities and consistent patterns emerge. Based on analysis of 99 polygonal tessellations with a wide variety of physical origins, this work demonstrates the ubiquity of an exponential distribution in the squared norm of the deformation tensor E2, which directly leads to the ubiquitous presence of gamma distributions in the polygon aspect ratio, as recently demonstrated by Atia et al. [Nat. Phys. 14, 613 (2018)]. In turn an analytical approach is developed to illustrate its origin. E2 relates to most energy forms, and its Boltzmann-like feature allows the definition of a pseudo temperature that promises utility in a thermodynamic ensemble framework.