E2 distribution and statistical regularity in polygonal planar tessellations

Abstract

From solar supergranulation to salt flat in Bolivia, from veins on leaves to cells on Drosophila wing discs, polygon-based networks exhibit great complexities, yet similarities persist and statistical distributions can be remarkably consistent. Based on analysis of 99 polygonal tessellations of 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 polygon aspect ratio. The $E^{2}$ distribution in turn arises as a $\chi^{2}$-distribution, and an analytical framework is developed to compute its statistics. $E^{2}$ is closely related to many energy forms, and its Boltzmann-like feature allows the definition of a pseudo-temperature. Together with normality in other key variables such as vertex displacement, this work reveals regularities universally present in all systems alike