Just today on numberphile channel they showed how to pass circle through smaller square hole by bending it in higher (3rd) dimension: https://m.youtube.com/watch?v=AvFNCNOyZeE
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Counterintuitive Properties of High Dimensional Space
11–20 of 68 posts
Re: Counterintuitive Properties of High Dimensional Space
#12from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…
Would it be counter intuitive though, that the ratio increases until dimension 5 (I think) then starts decreasing?
Re: Counterintuitive Properties of High Dimensional Space
#13Anyone interested in practical consequences of these counterintuitive properties should look up https://en.wikipedia.org/wiki/Curse_of_dimensionality for how it impacts things like machine learning where we naturally are working with many dimensions.
Re: Counterintuitive Properties of High Dimensional Space
#14from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…
Would it be counter intuitive though, that the ratio increases until dimension 5 (I think) then starts decreasing?
It's a matter of one's individual intuition, but said ratio is a non-linear function, and non-linear functions most often are non-monotonous.
Re: Counterintuitive Properties of High Dimensional Space
#15Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?
Re: Counterintuitive Properties of High Dimensional Space
#16from the article: The volume of the unit d-sphere goes to 0 as d grows! A high dimensional unit sphere encloses almost no volume! Maybe I'm nitpicking, but this interpretation is not accurate IMO. Volume of N-dimensional sphere is measured is different units than that of (N-1)-dimensional sphere. E.g. one is m^5, and another is m^4. Comparing values measured in different units is not the best idea. It would be better…
The ‘pithy’ version of this idea is that most of a high-dimensional orange is the peel.
It’s from this excellent article by Pedro Domingos: https://homes.cs.washington.edu/~pedrod/papers/cacm12.pdf
Re: Counterintuitive Properties of High Dimensional Space
#17Earlier quoted context omitted.
Would it be counter intuitive though, that the ratio increases until dimension 5 (I think) then starts decreasing?
It's a matter of one's individual intuition, but said ratio is a non-linear function, and non-linear functions most often are non-monotonous.
Except exponential, log, log-linear, sqrt, the time or space complexity of pretty much every problem in CS...
Re: Counterintuitive Properties of High Dimensional Space
#18Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?
We're using volume the whole time. Area is just the two-dimensional equivalent of volume. The two-dimensional equivalent of surface area would be perimeter, which is relatively rarely used.
Re: Counterintuitive Properties of High Dimensional Space
#19Why do we extrapolate 4d based on 3D volume, and not surface area? For 2D->3D we use area. Why would we then use volume in 3D->4D when surface area is an option?
Same reason we don't extrapolate 2D->3D based on circumference. You need all the information of the (n)D thing, plus a new length, to get to the (n+1)D thing.
Re: Counterintuitive Properties of High Dimensional Space
#20This was the first application of the Monte Carlo numerical method I've done in college. Take an N-dimensional cube with volume = 1. Inscribe an N-sphere in it. Calculate the volume of the N-sphere as a function of N. I was too lazy to do the strict proof, so I sprayed it with lots of Monte Carlo bullets. It's like a page of code in any language. It turns out, as the article says, the volume of the N-sphere keeps get…
John D. Cook has a couple of blog posts about non-intuitiveness of high-dimension geometry [1][2] and this article [3] expands on the very observation that you described.
Quote: "In his article “An Adventure in the Nth Dimension,” Brian Hayes explores how in high dimensions, balls have surprisingly little volume. As the dimension n increases, the volume of a ball of radius 1 increases until n = 5. Then for larger n the volume steadily decreases."
[1] https://www.johndcook.com/blog/2017/07/13/concentration_of_m...
[2] https://www.johndcook.com/blog/2017/07/19/corners-stick-out-...
[3] https://www.johndcook.com/blog/2012/10/23/dimension-5-isnt-s...