The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing
- Published
- Accepted
- Subject Areas
- Computational Biology, Mathematical Biology, Computational Science
- Keywords
- Voronoi diagram, Lewis’s law, Aboav-Weaire’s law, ellipse packing, von Neumann-Mullins’s law, 2D topology
- Copyright
- © 2019 Xu
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Preprints) and either DOI or URL of the article must be cited.
- Cite this article
- 2019. The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing. PeerJ Preprints 7:e27797v2 https://doi.org/10.7287/peerj.preprints.27797v2
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.
Author Comment
A few grammar and spelling mistakes were corrected, and the manuscript was formatted for submission.