On Tuesday, August 15, 2017 4:27:20 PM EDT Raul Miller wrote: > It loses accuracy somewhere between n=150 and n=200. > > Keep in mind though that the dimensions of these "volumes" are not > comparable.
That's a good point. Observing alone that the volumes of the spheres increase and then decrease as n increases is not the whole story to the dimensional weirdness. That property depends on the radius. Even the volumes of n-cubes with side length less than 1 decrease with n as well. But it is evident from the formula that the ratio of volumes of an n-sphere to its enclosing n-cube depend only on n and not the radius or side length. That fact that this ratio is strictly decreasing tells the story of the vanishing sphere. 0.25 0.5 1 2 (sphvol % (^~ +:)~)"0 1 i.6 1 1 0.785398 0.523599 0.308425 0.164493 1 1 0.785398 0.523599 0.308425 0.164493 1 1 0.785398 0.523599 0.308425 0.164493 1 1 0.785398 0.523599 0.308425 0.164493 But does it really get smaller? Check out 28:18 in the video Vijay posted: https://youtu.be/uU_Q2a0S0zI?list=PL2FF649D0C4407B30&t=1698 And we are not even considering fractional dimensions. ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
