ABSTRACT The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve these formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordering the seed locations of a regular hexagonal 2D Voronoi diagram, and analysed the cell topology based on ellipse packing. We then derived and verified the new formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of the edge number is 3. In addition, we derived a new formula for the von Neumann-Mullins law based on the improved Aboav-Weaire’s law. Our results suggest that the cell area, local neighbour relationship, and cell-growth rate are closely linked to each other, and primarily shaped by the effect of deformation from circle to ellipse, and less influenced by the global edge distribution.

中文翻译：

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基于椭圆堆积的二维路易斯定律和阿波夫-韦尔定律的几何公式

摘要 二维 (2D) Lewis 定律和 Aboav-Weaire 定律是从经验观察得出的两个简单公式。已经进行了许多尝试来改进这些配方。在这项研究中，我们通过随机排列正六边形二维 Voronoi 图的种子位置来模拟一系列 Voronoi 图，并基于椭圆堆积分析细胞拓扑。然后我们推导出并验证了刘易斯定律和 Aboav-Weaire 定律的新公式。具体来说，我们发现边数的二阶矩的上限为3。此外，我们基于改进的Aboav-Weaire 定律推导出了von Neumann-Mullins 定律的新公式。我们的结果表明细胞面积、局部邻居关系和细胞生长率彼此密切相关，