  By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a
planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is
developed on the basis of earlier theoretical work; it exploits, in particular,
the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have
between four and six significant digits. Accurate n dependent averages, second
moments, and variances are obtained for the cell area and the cell perimeter.
The numerical large n behavior of these quantities is analyzed in terms of
asymptotic power series in 1/n. Snapshots are shown of typical occurrences of
extremely rare events implicating cells of up to n=1600 sides embedded in an
ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic
features of such many-sided cells and their immediate environment. Their
relevance for observable properties is stressed.
