Random Polygons From Dirichlet Distributions And Extrapolation Estimations Of ?

The classical Archimedean approximation of?uses the semiperimeter of regular poly-gons inscribed in or circumscribed about a unit circle in R~2.With the doubling of the number of sides of these regular polygons,Archimedes discovered his famous harmonic-geometric-mean relations satisfied by the semiperimeter of the regular n-sided polygons and regular 2n-sided polygons.Through these recurrence relations,he calculated these quantities for n=6,12,24,48,96 and obtained the famous estimates 223/71