5 ms·

This is one of my favorite counterintuitive math results. > Volume of an n-ball tends to a limiting value of 0 as n goes to infinity Another way to say this i

by wging 4y ago

This is one of my favorite counterintuitive math results. > Volume of an n-ball tends to a limiting value of 0 as n goes to infinity Another way to say this is that if you inscribe a circle in a square, or a sphere in a cube, or in general an n-ball inside an n-cube, then in higher dimensions the n-ball takes up almost none of the space of the n-cube.

an1sotropy 4y ago
that the R=1 volume peaks at N=5 is not something I'll ever have intuition for.
- mturmon 4y ago
  Because the location of the peak falls at a different dimension depending on the underlying units (i.e., the radius), I don’t think the numerical location of the peak is fundamental. So maybe it’s best that you don’t have intuition for it?
  - dmurray 4y ago
    The ratio of the volume of the sphere to the volume of the enclosing cube (the proportion of the box that it takes up, or the probability that a randomly chosen point inside the box is in the ball) is dimensionless, though. It's not dependent on the size of the cube.
    - mturmon 4y ago
      You can extract dimensionless quantities, but depending on the normalization, the peak will land in different "n" values. E.g., normalize by the enclosing cube (sides of length 2R) versus the enclosed cube. So I don't think there's anything special about the turnover point.
- stadia42 4y ago
  The answers to this question provide at least some intuition: https://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5 https://math.stackexchange.com/questions/15656/volumes-of-n-....
VLM 4y ago
I've always thought that "density at the edges" would have interesting implications for moment of inertia. Euler's column formula depends on moment of inertia so that would seem to imply a "Flatland-but-in-11D-instead-of-2D" would have significant mechanical engineering problems compared to our 3-D world. I think macro-scale higher dimensions as a hard sci fi topic would also have weird aerodynamic and heat exchanger effects. How would the ratio of radiative cooling to convective cooling vary if the world had 10,11,100,100000000 dimensions instead of 3? Intuitively if we had 4 dimensions of space I think automotive-type radiators could be quite a bit smaller for a given heat exchange?

by wging 4y ago

an1sotropy 4y ago

that the R=1 volume peaks at N=5 is not something I'll ever have intuition for.

mturmon 4y ago
Because the location of the peak falls at a different dimension depending on the underlying units (i.e., the radius), I don’t think the numerical location of the peak is fundamental. So maybe it’s best that you don’t have intuition for it?
- dmurray 4y ago
  The ratio of the volume of the sphere to the volume of the enclosing cube (the proportion of the box that it takes up, or the probability that a randomly chosen point inside the box is in the ball) is dimensionless, though. It's not dependent on the size of the cube.
  - mturmon 4y ago
    You can extract dimensionless quantities, but depending on the normalization, the peak will land in different "n" values. E.g., normalize by the enclosing cube (sides of length 2R) versus the enclosed cube. So I don't think there's anything special about the turnover point.
stadia42 4y ago
The answers to this question provide at least some intuition: https://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5 https://math.stackexchange.com/questions/15656/volumes-of-n-....

VLM 4y ago

I've always thought that "density at the edges" would have interesting implications for moment of inertia. Euler's column formula depends on moment of inertia so that would seem to imply a "Flatland-but-in-11D-instead-of-2D" would have significant mechanical engineering problems compared to our 3-D world. I think macro-scale higher dimensions as a hard sci fi topic would also have weird aerodynamic and heat exchanger effects. How would the ratio of radiative cooling to convective cooling vary if the world had 10,11,100,100000000 dimensions instead of 3? Intuitively if we had 4 dimensions of space I think automotive-type radiators could be quite a bit smaller for a given heat exchange?