Something I worked on in my PhD was analyzing high dimensional bodies via their "sections."
Here's the Busemann Petty Problem:
Given two origin symmetric convex bodies K and L in n dimensions. Suppose for every linear hyperplane A (passing through the origin) we have vol_{n-1}(K intersect A) \leq vol_{n-1}(L intersect A).
Is it true that vol_{n}(K) < vol_{n}(L)?
[Here vol_k should be thought of as length when k = 1, area when k = 2, and volume in the traditional sense in k = 3.... generalizes quite well to arbitrary dimensions. And sections are these quantities L (resp. K) intersect A]
Turns out the answer is NO! In n \geq 10, it can be explained with the simple examples of K and L being the unit volume (vol_n) cube and a euclidean ball of volume (vol_n) slightly less 1 respectively. Comes from Keith Ball who, in his PhD thesis, established that {n-1}-section volume of the unit volume cube lies in [1, \sqrt(2)]. However for the euclidean ball of unit volume the section volume is at least sqrt(2). So you can start with the unit volume ball, decrease its radius infinitesimally so (the n-1 section volume falls less than the n-volume does) and generate a clear counterexample.
What this looks like is a ball with volume less than a cube but section volume seemingly leaks out of the faces of the cube. So a "spikey ball," if you may.
Not at all, the proof was a very elegant argument involving fourier transforms and an integral estimate going back to the study of controlling random walks (Khintchine's inequality). I say elegant in the manner of it being enviably so -- a proof a beginning graduate student could follow while capturing a fundamental, easy to state fact.
This work does however situate itself in/adjacent to that broad space of Brunn-Minkowski theory.
A sphere of course have sum of the squares of all points equal to one. So in high dimensions, it’s clear that most points have to be close to 0. Hence the “spikes”
Yes. I appreciate the article because it gets people to think about high dimensional geometry. But I think the right takeaway is that in high dimensions, cubes have huge volume compared to the enclosed sphere. So the sphere is not spikey at all. Instead realize that the diagonals and volume of the cube get very large compared to the sphere.
I think it's becoming obvious we need better metaphors for high-dimensional spaces than "it's like geometry except not at all".
At the end of it all, we have a big list of numbers (a vector) where each position in the list (component/dimension) implies a specific "meaning" that we don't know. We also have a variety of well-known mathematical operations we can do on those lists, the effects of which may depend on the number of positions present (the dimensionality of the vector space).
The challenge would be to find a good intuitive model to explain those effects (and ideally a way to visualise the lists that preserves the effects). Saying "it's an 1800 dimensional sphere" satisfies neither of those properties: You cannot visualise it and even if you want to think about it theoretically, it has none of the intuitive properties of a 2D or 3D sphere.
This video (I know sorry) will help out with dimensionality in many computing problems, which isn't the concept here, or what many people think of infinite or high dimensionality.
I think the effect is almost even more pronounced if you go in the other direction and look at a 1D space, i.e. the real line.
Then both a sphere with radius 1 and a box with side length 2 will become the same interval, with no difference.
In the OP setup, the middle sphere would vanish completely and (modulo some annoyances at the interval boundaries) become either the empty set or a single-element set, in either case having both radius and volume of zero.
The real brain burner is, neither. A sphere isn't spikey at all, actually. It's more than we feel like it should be. However, take that n-dimensional bounding cube and start rotating it, and it'll still contact the n-sphere in the same place, with exactly the same characteristics, but now on different points, even ones that were previously made of nothing but small numbers.
The sphere is "spiky" but you can't actually "rotate" the spikes because they aren't actually spikes. They're an artifact of us poor 3D-ers trying to understand higher dimensions.
"Unfortunately", 3 isn't very many dimensions, and in many cases we're really limited to 2 (our field of vision is more-or-less two, our depth perception is certainly not a direct apprehension of a third dimension but a bit of extra hinting on a fundamentally 2D view of the world), and that ends up not being a good view on to higher dimensions. I think even a 10D being would have less trouble imagining an 11th; they may still not be able to "visualize" it but with a larger sampling of what happens as you increase the dimension size they would be less fooled by artifacts of 0, 1, and 2 the way we are. We do not get very good generalization at 3, things are still dominated by special cases and small exponents.
(Of course, those "special cases" are part of why we are here in 3-space. Also, higher dimensions cause the general problem that the "body" of the life form has an exponentially increasing amount of space in their immediate vicinity where things may affect them, and that inevitably must be growing more slowly than the computing power of the life form contained within itself even though that may seem to be growing quickly to us. Even down here in 3-space we can be blindsided by things; a 10D being would never be able to keep track of their surroundings like we take for granted.)
You're on the right track. My comment alludes to sections, but there's a lot of essentially analogous math that explains the phenomena via shadows/"projections."
I love the counter-intuition of high-dimensional spaces, seems to be making the rounds on my feeds these days.
One of the harder generalizations to develop intuition for is the fact that the measure of a d-sphere tends to 0 as d approaches infinity, even though for all d = 0, 1, 2, 3 that our meager brains can visualize, the opposite is true! Geometry goes crazy.
Readit News
Here's the Busemann Petty Problem:
Given two origin symmetric convex bodies K and L in n dimensions. Suppose for every linear hyperplane A (passing through the origin) we have vol_{n-1}(K intersect A) \leq vol_{n-1}(L intersect A).
Is it true that vol_{n}(K) < vol_{n}(L)?
[Here vol_k should be thought of as length when k = 1, area when k = 2, and volume in the traditional sense in k = 3.... generalizes quite well to arbitrary dimensions. And sections are these quantities L (resp. K) intersect A]
Turns out the answer is NO! In n \geq 10, it can be explained with the simple examples of K and L being the unit volume (vol_n) cube and a euclidean ball of volume (vol_n) slightly less 1 respectively. Comes from Keith Ball who, in his PhD thesis, established that {n-1}-section volume of the unit volume cube lies in [1, \sqrt(2)]. However for the euclidean ball of unit volume the section volume is at least sqrt(2). So you can start with the unit volume ball, decrease its radius infinitesimally so (the n-1 section volume falls less than the n-volume does) and generate a clear counterexample.
What this looks like is a ball with volume less than a cube but section volume seemingly leaks out of the faces of the cube. So a "spikey ball," if you may.
This work does however situate itself in/adjacent to that broad space of Brunn-Minkowski theory.
Boxes are more intuitive
At the end of it all, we have a big list of numbers (a vector) where each position in the list (component/dimension) implies a specific "meaning" that we don't know. We also have a variety of well-known mathematical operations we can do on those lists, the effects of which may depend on the number of positions present (the dimensionality of the vector space).
The challenge would be to find a good intuitive model to explain those effects (and ideally a way to visualise the lists that preserves the effects). Saying "it's an 1800 dimensional sphere" satisfies neither of those properties: You cannot visualise it and even if you want to think about it theoretically, it has none of the intuitive properties of a 2D or 3D sphere.
It is worth your time IMHO.
https://youtu.be/q8gng_2gn70
Then both a sphere with radius 1 and a box with side length 2 will become the same interval, with no difference.
In the OP setup, the middle sphere would vanish completely and (modulo some annoyances at the interval boundaries) become either the empty set or a single-element set, in either case having both radius and volume of zero.
The sphere is "spiky" but you can't actually "rotate" the spikes because they aren't actually spikes. They're an artifact of us poor 3D-ers trying to understand higher dimensions.
"Unfortunately", 3 isn't very many dimensions, and in many cases we're really limited to 2 (our field of vision is more-or-less two, our depth perception is certainly not a direct apprehension of a third dimension but a bit of extra hinting on a fundamentally 2D view of the world), and that ends up not being a good view on to higher dimensions. I think even a 10D being would have less trouble imagining an 11th; they may still not be able to "visualize" it but with a larger sampling of what happens as you increase the dimension size they would be less fooled by artifacts of 0, 1, and 2 the way we are. We do not get very good generalization at 3, things are still dominated by special cases and small exponents.
(Of course, those "special cases" are part of why we are here in 3-space. Also, higher dimensions cause the general problem that the "body" of the life form has an exponentially increasing amount of space in their immediate vicinity where things may affect them, and that inevitably must be growing more slowly than the computing power of the life form contained within itself even though that may seem to be growing quickly to us. Even down here in 3-space we can be blindsided by things; a 10D being would never be able to keep track of their surroundings like we take for granted.)
One of the harder generalizations to develop intuition for is the fact that the measure of a d-sphere tends to 0 as d approaches infinity, even though for all d = 0, 1, 2, 3 that our meager brains can visualize, the opposite is true! Geometry goes crazy.
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