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I saw something quite similar that that blew my mind on high dimensionality (eg, on order of 10^23 dimensions). At low dimensions, if you take a sphere of radi

I saw something quite similar that that blew my mind on high dimensionality (eg, on order of 10^23 dimensions). At low dimensions, if you take a sphere of radius 1, and another sphere of radius 0.999999999999999 or so at the same origin, only a tiny fraction of the outer sphere's volume exists in the 'shell' between the two spheres. At very high dimensions, all of the interior volume is in the spherical shell. The fraction of volume in the interior becomes insignificantly infinitesimally small. Basically - at sufficiently-high dimensionality, a point chosen at random within the volume of an n-sphere will be on the surface.

Retric 10y ago
There is zero real difference, if you pick a large enough number of points you are likely to get within any arbitrary distance from the origin. It's just continuing the same trend from 1d to 2d an beyond.
nabla9 10y ago
There are so many weird things. * The volume of unit-radius sphere goes to zero as the dimension increases. * All the mass of hypersphere is near the equator (d-1 dimensional disk with thickness ε). * Alternatively almost all volume is close to surface like you said. * Almost all the surface area of a high-dimensional sphere is near the equator. * Distribution of distances between uniformly at random points concentrates around average distance as dimensionality grows. * 0 centered unit cube and unit sphere in multiple dimensions: vertices of the cube escape far away from the sphere while the face centers of the cube stay at the distance of 1/2 from the center. * Probability mass of Gaussians in multiple dimensions spreads all over the place.
- jacobolus 10y ago
  The unit of “hypervolume” is a hypercube. If you used a simplex as your unit instead, you’d have the hypervolume of a unit n-sphere go to infinity as n goes to infinity.
- kwhitefoot 10y ago
  > Almost all the surface area of a high-dimensional sphere is near the equator. Three dimensional spheres are invariant under rotation about any axis and the axes are not properties of the sphere. Equators are surely not properties of spheres. So how does a hypersphere have an equator? Or, there is a different meaning for some of these words than the ones I am used to; I didn't study maths at a higher level than that needed for a BSc. Physics forty years ago.
  - nabla9 10y ago
    With 'equator' I mean the cross section that cuts the sphere into two equal sizes and goes trough the center. Obviously choosing the equator is totally arbitrary in symmetric object.

Retric 10y ago

There is zero real difference, if you pick a large enough number of points you are likely to get within any arbitrary distance from the origin. It's just continuing the same trend from 1d to 2d an beyond.

nabla9 10y ago

There are so many weird things. * The volume of unit-radius sphere goes to zero as the dimension increases. * All the mass of hypersphere is near the equator (d-1 dimensional disk with thickness ε). * Alternatively almost all volume is close to surface like you said. * Almost all the surface area of a high-dimensional sphere is near the equator. * Distribution of distances between uniformly at random points concentrates around average distance as dimensionality grows. * 0 centered unit cube and unit sphere in multiple dimensions: vertices of the cube escape far away from the sphere while the face centers of the cube stay at the distance of 1/2 from the center. * Probability mass of Gaussians in multiple dimensions spreads all over the place.

jacobolus 10y ago
The unit of “hypervolume” is a hypercube. If you used a simplex as your unit instead, you’d have the hypervolume of a unit n-sphere go to infinity as n goes to infinity.
kwhitefoot 10y ago
> Almost all the surface area of a high-dimensional sphere is near the equator. Three dimensional spheres are invariant under rotation about any axis and the axes are not properties of the sphere. Equators are surely not properties of spheres. So how does a hypersphere have an equator? Or, there is a different meaning for some of these words than the ones I am used to; I didn't study maths at a higher level than that needed for a BSc. Physics forty years ago.
- nabla9 10y ago
  With 'equator' I mean the cross section that cuts the sphere into two equal sizes and goes trough the center. Obviously choosing the equator is totally arbitrary in symmetric object.