[Jprogramming] "n-volume" of an "n-sphere"

[Thomas Costigliola]

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.