Planar Voronoi cells and the failure of Aboav's law

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 $m\_n=4+3(\pi /n{)}^{-1/2}+...$ 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.