E2 distribution and statistical regularity in polygonal planar tessellations

• Published in: arXiv.org 2020-11
• Main Authors: Li, Ran, Ibar, Consuelo, Zhou, Zhenru, Moazzeni, Seyedsajad, Irvine, Kenneth D, Liu, Liping, Lin, Hao
• Format: Article
• Language: English
• Subjects: Aspect ratio; Flats (landforms); Fruit flies; Mapping; Marangoni convection; Polygons; Probability distribution functions; Statistical analysis; Statistical distributions; Tensors
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2002.11166

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