On Friday 08 December 2006 18:19, Daniel Asimov wrote:

The regular n-simplex delta_n(S_n) of side S_n is the convex span of n+1 points in n-space any two of which are a distance of S_n apart.

PUZZLE: Assume S_n = K*n for some constant K > 0. Define

f(n) = the n-volume of delta_n(S_n)

Given that f(n) approaches a finite limit L > 0 as n -> oo, find L.

I have a hideously mundane solution. I bet there's a better one. [... blank lines to avoid giving things away ...] Let h(n) := the height of a unit n-simplex delta_n(1). Then by Pythagoras we easily get h(n)^2 = 1 - ((n-1)h(n-1)/n)^2. Let k(n) := (n.h(n))^2; then k(n) = n^2-k(n-1), whence easily k(n) = n(n+1)/2 and h(n) = sqrt((n+1)/2n). Now the n-volume V(n) of delta_n(1) satisfies V(n) = h(n)/n V(n-1), so (iterating and substituting the formula above) V(n) = sqrt(n+1) / [n! 2^((n+1)/2)] so by Stirling V(n) ~ 1 / [(sqrt2/e)^n n^n 2sqrt(pi)]. Thus the volume of delta_n(Kn) ~ (Ke/sqrt2)^n / 2sqrt(pi), so we must take K = sqrt2/e yielding a limit L = 1/2sqrt(pi). I've probably dropped some factors of 2 and pi there :-). -- g