E^{2} and gamma distributions in polygonal networks

Abstract

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 E^{2}, 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)10.1038/s41567-018-0089-9]. In turn an analytical approach is developed to illustrate its origin. E^{2} relates to most energy forms, and its Boltzmann-like feature allows the definition of a pseudotemperature that promises utility in a thermodynamic ensemble framework.