Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis’s law and Aboav-Weaire’s law

Abstract

Background Lewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear. Methods We used R software with the Conicfit package to fit ellipses based on the geometric parameters of polygonal cells of ten different kinds of natural and artificial 2D structures. Results Our results indicated that the cells could be classified as an ellipse’s inscribed polygon (EIP) and that they tended to form the ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law. Conclusions Ellipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.

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“…1F). In this study, we confirmed that 2 could describe the deformation degree from EMIP to EIP which proposed by Xu (2019). Based on our data, we found the following equation…”

“…The coordinates of vertices, , and real (measured) area ( ) of each cell were extracted for the following analysis. For each polygonal cell, we used the R software (version 3.5.3) with the Conicfit package to fit an ellipse based on the coordinates of vertices (Chernov et al 2014;Xu 2019). The area of the maximal inscribed polygon of the fitted ellipse ( ) was calculated as = 0.5 (2 / ) (Su 1987).…”

“…Besides, the Weaire's sum rule suggested 2 = 〈 〉 − 36 = 〈 2 〉 − 36, which indicates that 2 ≥ 0 and 2 will increase with the edge range. Xu (2019) found that is actually a variance and equals to the ratio of major axis to minor axis of the fitted ellipse. Besides, the and 2 describe the deformation degrees from circle to ellipse, and from EMIP to EIP, respectively.…”

“…Two empirical laws, the Lewis's law and the Aboav-Weaire's law, were used to describe the relationships between the edge number ( ) and the cell area (A), and between and the average edge number ( ) of neighbor cells of the -edged cell, respectively (Aboav 1970;Lewis 1926;Lewis 1928;Weaire 1974;Weaire & Rivier 1984). Recently, based on investigations on ten different kinds of natural and artificial 2D materials, Xu (2019) found that the cells can be classified as an ellipse's inscribed polygon (EIP) and tended to form the ellipse's maximal inscribed polygon (EMIP). This phenomena was named as ellipse packing, which is a short-range order shaped the 2D topology by working together with the other short-range order, the trivalent vertices.…”

“…However, in the above study, the basic geometric data of ten kinds of 2D structures, such as coordinates of vertices, edge number, and cell area, were derived from the images (Xu 2019). This kind of data collection may affect the analysis, for example, it is very difficult to separate points and very short edges (Xu et al 2017).…”