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By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of aplanar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm isdeveloped 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 havebetween four and six significant digits. Accurate n dependent averages, secondmoments, and variances are obtained for the cell area and the cell perimeter.The numerical large n behavior of these quantities is analyzed in terms ofasymptotic power series in 1/n. Snapshots are shown of typical occurrences ofextremely rare events implicating cells of up to n=1600 sides embedded in anordinary Poisson-Voronoi diagram. We reveal and discuss the characteristicfeatures of such many-sided cells and their immediate environment. Theirrelevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table

New Monte Carlo method for planar Poisson–Voronoi cells