[math-fun] Strange behavior of volume of unit sphere as function of dimension

Let S^d denote the unit sphere of dimension d. According to my calculations, if vol : R —> R is the volume function of the unit sphere (the surface of the (d+1)-dimensional unit ball), then for d in the interval J J = [-10, -2] (even a bit larger), the volume function satisfies: -0.185 < vol(d) < 0.135 So, the range from minimum to maximum of vol over this interval of length = 8 satisfies max( vol(d) ) - min( vol(d) ) < .351 d in J d in J . Outside of this range, vol(d) moves quite fast; its average absolute derivative |vol'(d)| = d vol(d)/dd is pretty darn high (eyeballing). I wonder what accounts for this odd phenomenon. —Dan is less than 1/3.