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

Figure 1

(a)–(h) Examples of randomized polygonal networks. (a) Land cracks due to desiccation [2]. (b) Ice wedges from northern Canada [3]. (c) Veins on a Ficus lyrata leaf [5]. (d) Desiccation pattern of an ancient lake on Mars (HiRISE image: PSP_002140_2025 [26], credit to NASA/JPL/University of Arizona). (e) A snapshot of Salar de Uyuni in Bolivia, the world's largest salt flat [4]. (f) Supergranulation on the solar surface [1]. (g) Plated MDCK cells. (h) A processed image of a developing Drosophila wing disk with the aspect ratio of cells color mapped. (i) A regular hexagon ${\mathcal{P}}_R$ (blue, with vertices ${\mathbf{x}}_j$), a deformed hexagon $\mathcal{P}$ (red, ${\mathbf{y}}_j$), and a uniform deformation approximation ${\mathcal{P}}^{\prime }$ (black dashed outline, ${\mathbf{y}}_j^{\prime }$).

Figure 2

Universality in strain and aspect ratio distributions. The left column shows the PDFs of ${\left|\mathbf{E}\right|}^2$, fitted with an exponential form $exp(-\beta |\mathbf{E}{|}^2)$ to extract $\beta$. $\beta$ is then used in (6) to generate the theoretical curves in the middle column (dashed lines), in comparison to the aspect ratio (symbols). The coefficients of determination $R^2$ are shown. In the right column both data and theoretical curves from the middle column are normalized and fitted with a $k$-gamma function (7) (thick dashed lines). $k$ is extracted and shown in the legends. Data are from [28, 29] and this work.

Figure 3

(a) Overall correlation between $a_r$ and $T({\beta }^{-1}$) for all 99 data sets (symbols); the theoretical prediction (dashed line) is generated per (6). (b) A comparison between data and predicted temperature, $T_n$ and $T$ (“theory”). The inset shows that subensemble temperatures $T_n$ are quantitatively similar to the tessellation temperature $T$.

Figure 4

Normality of key variables; an asterisk (${}^{*}$) denotes normalization by its own standard deviation to compare with $N(0,1)$ (dark lines). (a) $\widehat{w}{}_{1,2}$ for all tessellations (198 profiles). (b) Normalized ${\left|\mathbf{E}\right|}^2$ follows a simple exponential distribution (99 profiles). (c) and (d) The normality of ${\widehat{u}}_k$ and ${\widehat{v}}_k$ for $n=6$, 1188 profiles each. (e) Initial displacement for two exemplary cases. (f) and (g) ${\widehat{u}}_k$ and ${\widehat{v}}_k$ asymptote toward normality. (g) Normalities of ${\widehat{w}}_{1,2}$ are well established.