Planar Voronoi cells and the failure of Aboav's law

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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, 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.