ABSTRACT

The equilibration of a trivalent polygonal network in two dimensions (2D) is a universal phenomenon in nature, but the underlying mathematical mechanism remains unclear. In this study, a relaxation algorithm based on a simple geometrical rule was developed to simulate the equilibration. The proposed algorithm was implemented using the Python programming language. The simulated relaxation changed the polygonal cell of the Voronoi network from an ellipse’s inscribed polygon toward the ellipse’s maximal inscribed polygon. Meanwhile, the Aboav-Weaire’s law, which describes the neighbouring relationship between cells, still holds statistically. The simulation results strongly supports the ellipse packing hypothesis that is proposed to explain the dynamic behaviours of a trivalent 2D structure. The simulated relaxation concentrated the interior angles around 120°, and increased the area and edge length of large cells but decreased the area and edge length of small cells. The area changes of cells with different edges due to relaxation could be described by the von Neumann-Mullins law. The results presented in this work can help to understand the mathematical mechanisms of the dynamic behaviours of trivalent 2D structures.

摘要

二维 (2D) 三价多边形网络的平衡是自然界中的普遍现象，但潜在的数学机制仍不清楚。在这项研究中，开发了一种基于简单几何规则的松弛算法来模拟平衡。所提出的算法是使用 Python 编程语言实现的。模拟松弛将 Voronoi 网络的多边形单元从椭圆的内接多边形更改为椭圆的最大内接多边形。同时，描述细胞之间相邻关系的 Aboav-Weaire 定律在统计上仍然成立。模拟结果强烈支持椭圆堆积假说，该假说旨在解释三价二维结构的动态行为。模拟弛豫将内角集中在120°左右，增大了大胞的面积和边长，而减小了小胞的面积和边长。具有不同边缘的单元由于弛豫引起的面积变化可以用冯诺依曼-穆林斯定律来描述。这项工作中提出的结果有助于理解三价二维结构动态行为的数学机制。