> 33. "...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent!
Counterintuitive facts about high dimensional geometry could get their own list. A side-1 cube in n dimensions has volume 1 of course, but a diameter-1 sphere inside it has volume approaching zero! The sphere is tangent to every one of the 2n faces, yet takes up almost none of the space inside the cube.
Note that the distance from the middle of any face of the cube to the opposite face is 1, yet the length of a diameter of the cube (corner to opposite corner) is sqrt(n).
Only sort of true. It doesn't make sense to compare n dimensional volume to n+1 dimensional volumes, so the limit of the volume of an n-sphere isn't meaningful. The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.
In particular, it's equally valid and frankly nicer to define the unit n-sphere to be volume 1 rather than the unit cube. Do that and we see that this statement is just saying that the n-cube grows in volume to infinity, which makes sense given the fact you point out that it contains points increasingly far from the origin.
I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
> The limit that does make sense is the ratio of volumes of n-sphere to an n-cube. That that goes to zero is maybe not so surprising.
This is why I start by recalling that the volume of the n-cube is always one, as the frame of reference. But I think people still find it surprising. Hard to tell, because...
> I have a hobby of turning surprising facts about the n-sphere into less surprising facts about the n-cube. So far I haven't met one that can't be 'fixed' by this strategy.
Hard to tell, because I don't find any of these facts surprising anymore -- would guess you're in the same boat!
Another good one is how you can fit exp(n) "almost-orthognal vectors" on the n-sphere.
> 33. "...if you flip fair coins to generate n-dimensional vectors (heads => 1, tails => -1) then the probability they're linearly independent is at least 1-(1/2 + o(n))^n. I.e., they're very very likely independent!
Counterintuitive facts about high dimensional geometry could get their own list. A side-1 cube in n dimensions has volume 1 of course, but a diameter-1 sphere inside it has volume approaching zero! The sphere is tangent to every one of the 2n faces, yet takes up almost none of the space inside the cube.
Note that the distance from the middle of any face of the cube to the opposite face is 1, yet the length of a diameter of the cube (corner to opposite corner) is sqrt(n).