<< Even more bizarre, IMHO, is the fact that in the volume formula for n dimensions, it isn't pi itself, but sqrt(pi), that is important.

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Not true. The "surface" (n-1)-measure of the (n-1)-sphere is S(n-1) = (2 pi^(n/2) / Gamma(n/2)) r^(n-1), and the "volume" n-measure of the n-ball is V(n) = (2/n) pi^(n/2) / Gamma(n/2)) r^n, but the sqrt(pi)'s that appear for n odd cancel in the numerator and denominator, so that all S(n-1)'s and V(n)'s for r = 1 are rational numbers times integer powers of pi. The power of pi is floor(n/2) for both S(n-1) and V(n), so we list the power of pi in parentheses after n, and just list the rational numbers in the lines below: n 0(0) 1(0) 2(1) 3(1) 4(2) 5(2) 6(3) 7(3) 8(4) ---------------------------------------------------- S(n-1) - 2 2 4 2 8/3 1 16/15 1/3 ---------------------------------------------------- V(n) 1 2 1 4/3 1/2 8/15 1/6 16/105 1/24 if my calculations are correct. It is amusing to note that the sum of the V(n) for all even n is e^pi (if one reasonably takes V(0) = 1, since the 0-ball is just a point and 0-dimensional measure just counts points). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele