Abstract
BackgroundLewis’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.MethodsWe 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.ResultsOur 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.ConclusionsEllipse 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.
Highlights
A two-dimensional (2D) plane can be tessellated by convex polygons
We found the average number of edges of P. haitanensis cells to be 6.0 ± 0.9 (1375 cells in 13 thalli were examined; Table 1), which was consistent with previous studies on P. haitanensis as well as studies on many other organisms and physical structures (Gibson et al, 2006; Sánchez-Gutiérrez et al, 2016; Weaire & Rivier, 1984; Xu et al, 2017)
This study found that polygonal cells of natural and artificial 2D structures were inclined to form ellipse’s maximal inscribed polygon (EMIP)
Summary
A two-dimensional (2D) plane can be tessellated by convex polygons. Scientists are interested in natural and artificial 2D structures that share the common feature that the coordination number of vertices (the number of edges meeting at a vertex) of polygonal cells always equals three. The cell topology of these 2D structures can be described according to three laws: Euler’s law, Lewis’s law, and Aboav-Weaire’s law (Weaire & Rivier, 1984). 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. The results of this study could be used to simulate the dynamics of cell topology during growth
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