Universal Topological Properties of Two-Dimensional Trivalent Cellular Patterns
Kwok Yip Szeto, Xiujun Fu, Wing Yim Tam · Physical Review Letters · 2002
Universal topological properties of two-dimensional trivalent cellular patterns are found from shell analysis of soap froth and computer-generated Voronoi diagrams. We introduce a cluster analysis based on the shell model and find the universal relation $\mathrm{ln}(a/{\ensuremath{\mu}}_{2})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}A+B\mathrm{ln}({\ensuremath{\mu}}_{2})$, with the generalized Aboav parameter $a$ and second moment of the number of cell edge distribution ${\ensuremath{\mu}}_{2}$. For the second, third, and fourth shells of the cluster, $A$ and $B$ are the same for all samples. Furthermore, $A$ is increasing with shell number while $B$ is a universal number, $\ensuremath{-}0.90$. For the first shell, the slope $B$ is the same for soap froths, but slightly different from Voronoi graphs.