[HN Gopher] Spikey Spheres (2010)
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Spikey Spheres (2010)
Author : alexmolas
Score : 33 points
Date : 2024-07-31 08:53 UTC (14 hours ago)
(HTM) web link (www.penzba.co.uk)
(TXT) w3m dump (www.penzba.co.uk)
| petters wrote:
| 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"
|
| Boxes are more intuitive
| figure8 wrote:
| 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.
| zardo wrote:
| The 'spikes' depend on an arbitrary choice of coordinate
| system.
| petters wrote:
| Yes, that is true
| dexwiz wrote:
| Is the sphere spikey or its shadow in lower dimensional space?
| Like how the shadow of a disk from the side would look like a
| long line.
| krnsll wrote:
| 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."
| jerf wrote:
| 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.)
| nyrikki wrote:
| 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.
|
| It is worth your time IMHO.
|
| https://youtu.be/q8gng_2gn70
| krnsll wrote:
| 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.
| selimthegrim wrote:
| Does Brunn-Minkowski get used here at all wrt (maybe mixed)
| volume?
| krnsll wrote:
| 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.
| xg15 wrote:
| 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.
| krukah wrote:
| 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.
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