Geometric formulas of Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

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.

KEYWORDS:

• Lewis’s law
• Aboav-Weaire’s law
• 2D topology
• ellipse packing
• von Neumann-Mullins law

Acknowledgements

The author thanks Professor Rolf Turner for the development of the R package deldir and his technical support. The author also thanks Miss Fangyu Guo and Ping Zhang for their assistance collecting the geometric cell data. Thanks are also due to my beloved wife, Huimin Cheng, for her support.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Kai Xu http://orcid.org/0000-0002-1341-1525

Additional information

Funding

This work was supported by the National Key Research and Development Program of China [grant number 2018YFD0900702].