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A geometry-based relaxation algorithm for equilibrating a trivalent polygonal network in two dimensions and its implications

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 in Python 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 neighboring relationship between cells, still holds statistically. The succeed of simulation strongly supports the ellipse packing hypothesis that was proposed to explain the dynamic behaviors of a trivalent 2D structure. The simulation results also showed that the edge of large cells tends to be shorter than edges of small cells, and vice versa. In addition, the relaxation increases the area and edge length of large cells, and it decreases the area and edge length of small cells. The pattern of changes in the area of different-edged cells due to relaxation is almost the same as the growth pattern described by the von-Neumann-Mullins law. The results presented in this work can help to understand the mathematical mechanisms of the dynamic behaviors of trivalent 2D structures.