The equilibration of  trivalent polygonal network in two dimensions (2D) is an universal phenomena in nature, however, the behind mathematical mechanism remains unclear. In this study, to simulate the equilibration, we developed a relaxation algorithm just based on one simple geometrical rule . The simulated relaxation changed the polygonal cell of  Voronoi network from  ellipse's inscribed polygon toward  ellipse's maximal inscribed polygon. Meanwhile, the Aboav-Weaire's law, which describe the neighboring relationship between cells, still holds statistically. The implementation of this algorithm strongly support the hypothesis Ellipse Packing which was proposed to explain the dynamic behaviors of  trivalent 2D structure. We also found that the edge of large cells tend to be shorter  , and vice versa. Besides , the relaxation increased the area and the edge length of large cells and vice versa . The pattern of changes on the area of different edged cells due to relaxation is almost the same compared with the growth pattern described by  von-Neumann-Mullins 's law. This work could help to understand the mathematical mechanisms of the dynamic behaviors of trivalent 2D structures.
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.
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