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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 the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis's law and Aboav-Weaire's law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins's law based on the new formula of the Aboav-Weaire's law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.