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by locuscoeruleus 1132 days ago
This was my immediate thought as well! Can someone more mathematical say if this is an unreasonable thought?
2 comments
j7ake 1132 days ago
As one increases more dimensions, it becomes increasingly likely that at least one of them will be an "extreme" value (i.e. point is at the edge of the hypercube ).

In the random() * random() * random() ... , once you draw at least one number close to 0 (chances will increase as you add more randoms), then your output collapses to a very small number. And in this visualization, 0 is shown as a point at the edge of the circle.

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tysam_and 1132 days ago
It depends upon what metric you use to measure it.

Of course with Euclidean it will as that seems to match our expectations for low dimensional space.

I'm not a mathematician so there is not much more that I could say on the matter at this juncture.

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j7ake 1132 days ago
The volume of an n-ball has been calculated for other non-Euclidean distances.

For p goes to infinity, the unit "ball" becomes a "cube" of two units wide per side.

https://en.wikipedia.org/wiki/Volume_of_an_n-ball#Balls_in_L...

https://www.johndcook.com/blog/2010/07/02/volumes-of-general...

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tysam_and 1132 days ago
Many thanks, I've been wondering about this one for quite a while. <3 :))))
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