Planar Voronoi cells and the failure of Aboav's law

Hendrik-Jan Hilhorst

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Planar Voronoi cells and the failure of Aboav's law

Hendrik-Jan Hilhorst

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Abstract

Aboav's law is a quantitative expression of the empirical fact that in planar cellular structures many-sided cells tend to have few-sided neighbors. This law is nonetheless violated in the most widely used model system, {\it viz.} the Poisson-Voronoi tessellation. We obtain the correct law for this model: Given an $n$-sided cell, any of its neighbors has on average $m\_n$ sides where $m\_n=4+3(π/n)^{-{1/2}}+...$ in the limit of large $n$. This expression is quite accurate also for nonasymptotic $n$ and we discuss its implications for the analysis of experimental data.

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Hendrik-Jan Hilhorst. 2005-09-15. Planar Voronoi cells and the failure of Aboav's law. https://arxiv.org/abs/cond-mat/0509409

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