  We consider the three-dimensional Poisson-Voronoi tessellation and study the
average facedness m_n of a cell known to neighbor an n-faced cell. Whereas
Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic
expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar
et al., based on a very large data set, now clearly rule out Aboav's law. In
this work we determine the numerical value of k_1 and compare the expansion to
the Monte Carlo data. The calculation of k_1 involves an auxiliary planar
cellular structure composed of circular arcs, that we will call the
Poisson-Moebius diagram. It is a special case of more general Moebius diagrams
(or multiplicatively weighted power diagrams) and is of interest for its own
sake. We obtain exact results for the total edge length per unit area, which is
a prerequisite for the coefficient k_1, and a few other quantities in this
diagram.
