[HN Gopher] A high-dimensional sphere spilling out of a high-dim...
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A high-dimensional sphere spilling out of a high-dimensional cube
Author : EvgeniyZh
Score : 204 points
Date : 2022-01-17 17:05 UTC (5 hours ago)
(HTM) web link (stanislavfort.github.io)
(TXT) w3m dump (stanislavfort.github.io)
| emsign wrote:
| There's a key difference between how we perceive and think about
| two dimensional space and 2+n dimensional space. If you want to
| seal in one geometric object with other geometric objects like in
| this example then in 2D it's sufficient to let the enclosing
| objects touch each other at their widest points, but in higher
| dimensions they have to touch each other at their narrower
| points, overlapping into each other in many cases. And in the
| first example which was the 2D example the outer objects touch
| each other at their outmost points. In 2D it looks like a full
| enclosure with no way for the inner circle to get through. But
| our brains assume that the higher dimension versions will be 100%
| sealing it off as well, because we saw it in the first example,
| right?
|
| This is an example of a misleading first example. What actually
| will happen and what you expect will happen are not the same,
| because you got tricked by the first example.
| smegsicle wrote:
| and you can see the inner circle is much smaller than the
| surrounding circles in 2d, the difference is a little less so
| in 3d? which would continue until the so-called 'inner' circle
| is friggn _huge_.
|
| I mean, everyone knows that everything is super far apart in
| higher dimensions, it surprises me that it takes up to 10D
| before he slips out-- perhaps more intuitive when i keep in
| mind that n-spheres are the most compact shape?
| scatters wrote:
| I don't think it's correct to see the n-sphere as poking out of
| the sides of the cube; the sphere has constant curvature so it
| has no protrusions. Rather, the n-cube gets more and more spiky
| so the constraining spheres get (relatively) smaller and further
| away from the centre, so constrain the central sphere less and
| less.
|
| The spikiness of n-cubes is apparent when you look at the (solid)
| angle at each vertex; it starts constant (1/2 pi radian in 2D,
| 1/2 pi steradian in 3D) but reduces thereafter at an increasing
| rate (1/8 pi^2, 1/12 pi^2, 1/64 pi^3, etc.).
| scythe wrote:
| The radius of the sphere, assuming the cube has side length 1,
| is (sqrt(D)-1)/2. It seems worthy of note that this radius
| becomes arbitrarily large.
| kgwgk wrote:
| And the distance from a vertex of the unit cube to the center
| is sqrt(D)/2, which also becomes arbitrarily large.
|
| https://www.math.ucdavis.edu/~strohmer/courses/180BigData/18.
| ..
|
| > The result, when projected in two dimensions no longer
| appears convex, however all hypercubes are convex. This is
| part of the strangeness of higher dimensions - hypercubes are
| both convex and "pointy."
| AnimalMuppet wrote:
| Isn't it more accurate to say that hypercubes are convex
| and _their projections_ are pointy?
| [deleted]
| kgwgk wrote:
| Yes. Hence the quotes, I guess. But hyperspheres are
| convex and their projections are also convex. So it's not
| really appropriate either to say that they are "spiky" as
| in https://news.ycombinator.com/item?id=29969613
| thaumasiotes wrote:
| > the (solid) angle at each vertex; it starts constant (1/2 pi
| radian in 2D, 1/2 pi steradian in 3D)
|
| Unstudied curiosity: steradians seem to be defined in terms of
| the point of a cone. But the corner of a cube is the
| intersection of several (for a cube, 3) planes, not a cone. How
| do you do the calculation of steradians?
| wlesieutre wrote:
| Cone is one scenario to think about, but what it really comes
| down to is the fraction of the full sphere that your solid
| angle includes.
|
| To make a 2D analogy, you can think of 2D angles as a portion
| of a circle represented as a fraction of 2pi radians. Cut it
| in half and you have pi radians, cut that in half (so a
| quarter of the circle) and you have pi/2 radians, or 90
| degrees.
|
| That 90 degrees "quarter of a circle" example is looking at
| it as the "point of a pie slice", but it's the same 90 degree
| 2D angle as you have in the corner of a square.
|
| You can look at the corner of a cube the same way. The full
| sphere of solid angle is 4pi, a hemisphere is 2pi. Now take
| that hemisphere and cut it into quarters (1/8ths of a
| sphere). Each of those quarters is pi/2 steradians, and the
| solid angle at the center is the same solid angle represented
| by 1/8th of the sphere is the same solid angle you have at
| the corner of a cube.
|
| Or to put it another way, you could pack the corners of 8
| cubes around a point and it would leave no empty gaps, so the
| corner of each cube is occupying 1/8th of a "full" 4pi
| steradians.
| thaumasiotes wrote:
| Thanks, that was helpful in thinking about it.
| shenberg wrote:
| The spiky view of n-spheres is due to all of their volume
| getting concentrate next to the axes - the proportion of volume
| which is less than e away from an axis to the entire volume of
| the sphere tends to 1 exponentially with n, which leads to the
| spiky visualization as we imagine 3d but with more axes.
| 6gvONxR4sf7o wrote:
| The n-cube is spiky. The n-sphere is kinda defined by its
| non-spikiness, because it is rotationally invariant.
| kgwgk wrote:
| But there is an alternative visualization where the sides of
| the unit (hyper)cube are deformed because the location of the
| vertices gets further and further away from the origin. Of
| course neither is a true representation of the
| hyperdimensional case.
| mgraczyk wrote:
| I don't think this is the right way to process this
| intuitively. When you use words like "volume getting
| concentrated" it sounds like there is some non-uniformity in
| the sphere, but the non-uniformity is really in our intuition
| about space.
|
| What's weird isn't the sphere, it's distance, and I think
| that's easier to process. Going from a (1d) sidewalk to a
| (2d) football field to a (3d) ocean, it's easier to see our
| intuitions about distances slowly breaking down.
| vba616 wrote:
| >it sounds like there is some non-uniformity in the sphere
|
| How I am understanding this is that the non-uniformity is
| in the cube, and I think it's very helpful in visualizing
| it.
|
| I can imagine in 3d, the centers of the sides of the cube
| being pulled in, so that it's kind of hyperbolic looking.
| sdwr wrote:
| Love this explanation, visualizing corner vs centre is
| intuitive in a way higher dimensions usually aren't.
| tgb wrote:
| I'm not certain I know what you mean, but it doesn't seem to
| work. If you mean to take epsilon neighborhoods in Euclidean
| norm of the axes, then I'm skeptical that that contains most
| of the mass of the sphere, since simulating random points on
| a thousand dimensional sphere doesn't seem to give points
| near these axes.
|
| If instead you mean most of the mass of the ball is in the
| points which have all but one coordinate within epsilon of
| zero, then the intuition doesn't follow. It's equally true
| that you get most of the mass considering just points with
| _all_ coordinates within epsilon. And for that you get that
| the mass of the ball is concentrated in an epsilon cube at
| the origin, which excludes exactly the spikes that you were
| basing the intuition on. The weird thing in this thought
| experiment is the epsilon cube not the sphere. For example
| the epsilon cube contains points much further than epsilon
| from the origin and so it 's maybe not surprising that it
| contains most of the sphere.
| trhway wrote:
| >since simulating random points on a thousand dimensional
| sphere doesn't seem to give points near these axes.
|
| that is kind of circular argument as "random" really
| depends on the density measure underlying the chosen
| sampling distribution.
| kgwgk wrote:
| If only there was some kind of natural density measure to
| use in an Euclidean space R^D...
| tgb wrote:
| It's no more ill defined than volume itself (and less so
| since it doesn't need an arbitrary scale).
| momenti wrote:
| https://news.ycombinator.com/item?id=3995615
| tgb wrote:
| Perhaps you can be more specific since I don't see the
| claim here being discussed there.
| ColinWright wrote:
| This phenomenon has been written up many times, and some of those
| have been submitted here previously, with some discussion. For
| those who might be interested to see those previous discussions,
| here are two of them:
|
| https://news.ycombinator.com/item?id=12998899
|
| https://news.ycombinator.com/item?id=3995615
|
| Some of the comments here were made in those discussions, but
| some of the comments on those discussions have not yet been made
| here.
| capableweb wrote:
| > Some of the comments here were made in those discussions, but
| some of the comments on those discussions have not yet been
| made here.
|
| Many of the missing comments from those linked submissions have
| also not been made here yet. Hope we'll see them soon.
| it_does_follow wrote:
| To add to this list: Richard Hamming includes a section on this
| in his n-dimensional spaces talk from "The Art of Doing Science
| and Engineering" lectures [0]. Stripe Press also recently re-
| published a beautiful copy of the print version of these
| lectures [1].
|
| 0. https://youtu.be/uU_Q2a0S0zI?t=1716
|
| 1. https://press.stripe.com/the-art-of-doing-science-and-
| engine...
| growt wrote:
| At university I initially chose math as a minor. I think it was
| this problem (without the pretty pictures) where I decided that
| math was not for me.
| rwmj wrote:
| Just think of the high dimension n-cube like a spiky sea urchin.
| It has 2^n spikes, and the spheres live in those spikes near the
| ends. The central sphere is large because it extends out to those
| spheres, extending outside the sea urchin's "body".
| feoren wrote:
| But ... it's not. It's not concave anywhere. If you draw a line
| from any point of the n-cube to any other point, it never
| passes outside the body of the cube. Perhaps your model gives
| better intuition in "curse of dimensionality" cases like this
| one, but it's clearly worse in other ways, right? It's simply
| not at all an accurate description of the shape.
| thadk wrote:
| Doesn't Alicia Boole Stott's ability indicate that solid
| intuitions are plausible though?
| https://www.askaboutireland.ie/reading-room/life-
| society/sci...
| feoren wrote:
| Maybe? I'm not claiming there's no way to have a good
| intuition about 4D space -- in fact articles like this make
| me want to figure out how to achieve such a thing. But it
| seems likely to me that even if your brain is somehow
| capable of visualizing 4D things, it would be just as weird
| to move to 5D as it is for normal people to move to 4D. Did
| Stott have any special intuition about 5+D? And we're
| talking about making that cognitive jump _five_ more times
| to get to 10D.
|
| However, it's clear the "starfish" intuition is simply not
| accurate. That's not what N-cubes look like. The point of
| this post is that we _should_ have cognitive dissonance
| when we try to think about 10-cubes, because it 's _weird_
| that (A) the "inner sphere" pokes out of a shape that is
| (B) convex everywhere. You can resolve the cognitive
| dissonance easily by simply ignoring or rejecting B --
| sure, it's not weird that such a sphere would poke out of a
| starfish. But _you are wrong_. It 's _not_ a starfish! It
| 's convex everywhere! So you can't say "why do y'all have
| cognitive dissonance about this?"
| sdwr wrote:
| It is accurate, just not completely accurate. You only
| get cognitive dissonance if you try to resolve it all the
| way.. stack multiple imperfect intuitions to approximate
| the real thing.
| wruza wrote:
| It depends on how you define "concaveness". A cube is
| concaver than a square _in a sense_. The travel from the
| center of a unit square to its side takes 0.5, and to its
| vertex ~0.7. For a cube, 0.5 and ~0.87. For a 100d cube, 0.5
| and 5. (For completeness, for 1d cube it 's 0.5, 0.5). Of
| course that is not a true concaveness, but it gives a nice
| sense of inside distances, especially with spheres, which are
| defined as equidistant.
|
| We are just used to project a cube in a way that prevents to
| see its linearly-spatial configuration (a projection messes
| up lengths), but if you preserve these lengths and "flatten"
| them instead, a cube will flatten out to a sort of a
| shuriken.
| function_seven wrote:
| > _It depends on how you define "concaveness". A cube is
| concaver than a square in a sense._
|
| Yup. Even a regular 3D cube (and 2D square) has concave
| faces if you're viewing it from a polar perspective. As I
| stand in the center of the cube, measuring distances from
| me to the surface, I'll see that measurement follow a
| concave pattern.
|
| Yeah, I know that's not the definition of concavity or
| whatever, but when relating a sphere to a cube, and trying
| to get an intuition of higher-dimensional spaces, I think
| it helps to look at it from the sphere's perspective rather
| than a Cartesian one.
| brummm wrote:
| There is a very clear definition of convexity, as can be
| seen here [1] on Wikipedia. Nothing to discuss about with
| regards to definition.
|
| [1] https://en.wikipedia.org/wiki/Convex_set
| wruza wrote:
| Maybe there is a better word to describe this idea?
| sdwr wrote:
| It works if you aren't rigid about your mental
| representations. Pull an inverse wittgenstein on the
| intuitions. Instead of stool, chair, recliner all being
| instances of the general category "chair" -- sea urchin,
| cube, dodecahedron are all partially accurate descriptions of
| the specific 10D cube.
| 6gvONxR4sf7o wrote:
| There's a certain way in which a cube "feels sharper" or
| "feels spikier" than a square. Trying to formalize that, you
| can compare the edge of a 3d box where two faces meet to the
| point where three faces meet. I'd rather step on the two-face
| edge than the three-face corner, and there's definitely a
| sense in which the cube is spikier.
|
| It seems reasonable to extend that same intuition to n-D
| sharpness/spikiness in an accurate way. Adding an extra
| "face" just chops more off making those vertices sharper and
| sharper, at least relative to the high dimensional space
| around it.
| feoren wrote:
| I think that's a great insight; I especially like comparing
| the sharpness of an two-face edge to a three-face corner;
| we could expect stepping on a seven-face corner to be
| slightly worse than stepping on a six-face. I think that
| "sharpness" idea surely must be related to this phenomenon.
| However, one should be careful not to let that "increasing
| sharpness" idea lead to the mental image of a concave
| shape, especially not a sea urchin. That would be a false
| resolution to this "paradox" and that image of a N-cube
| would lead to all sorts of other incorrect ideas, e.g.
| regarding where the volume of the shape is.
| 6gvONxR4sf7o wrote:
| Maybe the right intuition is a sea urchin that is, due to
| high dimensional unintuitive properties, still convex.
| It's almost entirely extremely pointy corners, and yet
| they're "magically" connected to each other in an
| entirely convex manner.
| fghorow wrote:
| Is there some obvious way (that I seem to be missing) to see that
| the centrally inscribed D-sphere _must_ touch all of the other
| spheres in high dimensions?
|
| That's probably a stupid question, but while that fact is
| intuitively obvious for D={2,3} -- as this problem tries to
| demonstrate -- higher dimensions are unintuitively WEIRD.
| SamReidHughes wrote:
| Symmetry.
| fghorow wrote:
| OK, Another reference is [1], that agrees with the result given
| by the OP.
|
| I'm not trying to do research here, I'm just boggling at the
| unintuitive result, and trying to see if there might be a flaw
| in the chain of logic. The fact that this is "well known" is
| enough to scare me off from barking up this particular tree.
|
| [1] https://www.math.wustl.edu/~feres/highdim
| nealabq wrote:
| You can argue by symmetry that if the central sphere touches
| one of the corner spheres, it must touch them all. And it must
| touch one because otherwise you would increase it's radius
| until it did.
| Dylan16807 wrote:
| Well we can calculate the touch point for one sphere, and we
| know that it would overlap that sphere if it was radius
| sqrt(D).
|
| And all the other spheres are simple symmetrical mirrors, so
| how could it not touch all of them at the same time it touches
| one? That should scale to an arbitrary number of dimensions,
| right?
| cantagi wrote:
| I also found it very weird, but here's my intuition.
|
| There are 2^k > 512 spheres stuck to eachother across k-d
| (pretend k=9). The line from the center to the point where the
| inner sphere touches one of outer spheres has to shortcut
| through all k dimensions to get from the center to the sphere.
|
| This distance has been massively inflated due to the number of
| dimensions. But the distance to the edge of the box hasn't been
| inflated - it's just constant, so the inner sphere breaks out.
| spenczar5 wrote:
| Substep 6 is not obvious to me. It is not clear that the point
| where the inner sphere and the space-filling spheres intersect
| _must_ be along the line from the center of the cube to the
| center of each sub-cube.
|
| To put it another way, it's not obvious to me that the point of
| contact between the inner pink circle and the outer black circle
| is along the green line.
| jjgreen wrote:
| Both spheres are symmetric around this line, so a point on
| intersection anywhere but on the line would give you a circle
| of intersection (and so a disk, by convexity) ...
| spenczar5 wrote:
| How do you know they are both symmetric around this line? The
| rest of your argument makes sense to me, yeah.
| unnah wrote:
| The spheres are symmetric around the line because the line
| goes through the center points of the spheres.
| hervature wrote:
| Not obvious for higher dimensions or even D=2? Surely you agree
| the center of both circles (regardless of dimension) occur
| along the line from the center of the cube to one of the
| corners. Therefore, if you just radially grow these spheres
| they must touch for the first time along this line. To be
| clear, this is nothing special about the cube, if you draw a
| line between the center of any two circles, the first time they
| touch will be somewhere along this line. In this case, the
| inner circle is obviously along the diagonal and it doesn't
| take much to see that the outer circles are as well by their
| construction. Therefore, the diagonal is the line that connects
| the centers.
| spenczar5 wrote:
| I don't agree that the center of both circles needs to lie
| along that line. The space-filling ones do by definition, but
| I don't see why the center one has to be on that line. It
| seems like it must in D=2, of course, but I couldn't prove
| that it must for D=9, or even that it is unique.
| ted_dunning wrote:
| The line goes from the center of the cube to the corner.
| The central sphere is concentric with the cube. Therefore
| the center of the central sphere is on the line from the
| center to the corner. In fact, it the center of that sphere
| is on any line from the center of the cube to anywhere.
| jeeceebees wrote:
| I think this is a property spheres. It seems to me that any two
| spheres that are touching have a straight line from one center
| to the other center exactly through the point of contact. Try
| thinking of just two spheres and adding more in step-by-step.
|
| Then the result follows because all the spheres are defined as
| centered on the cube/sub-cubes respectively.
| spenczar5 wrote:
| The inner sphere is not defined as centered on the cube; it
| is defined as touching all the other spheres.
|
| That said, there is a symmetry argument that if it were
| centered anywhere else, something is wrong. But that only
| works if there is only one _unique_ sphere that touches all
| the other spheres, which is also not obvious to me in higher
| dimensions.
| Dylan16807 wrote:
| You can go ahead and define it as centered on the cube.
| That still demonstrates the strange nature of high-
| dimensional spheres even if there wasn't a unique solution
| for touching all the other spheres.
| spenczar5 wrote:
| Aha! Right, this is pretty convincing to me. Thanks!
| hprotagonist wrote:
| relatedly, euclidean distance is a shitty metric in high
| dimension.
| AnimalMuppet wrote:
| How so? And, what is a _better_ metric, and why?
| hprotagonist wrote:
| cosine similarity's a lot better.
|
| https://stats.stackexchange.com/questions/99171/why-is-
| eucli...
| CrazyStat wrote:
| Cosine similarity is not a distance metric.
| contravariant wrote:
| Well it is on the unit sphere, but then it's equivalent
| to the euclidean metric...
| srean wrote:
| Do I spot a geometer here ? You are indeed right but its
| not something that is well known.
| anon_123g987 wrote:
| For example:
|
| _The Mahalanobis distance is a measure of the distance
| between a point P and a distribution D, introduced by P. C.
| Mahalanobis in 1936. It is a multi-dimensional generalization
| of the idea of measuring how many standard deviations away P
| is from the mean of D. This distance is zero for P at the
| mean of D and grows as P moves away from the mean along each
| principal component axis. If each of these axes is re-scaled
| to have unit variance, then the Mahalanobis distance
| corresponds to standard Euclidean distance in the transformed
| space. The Mahalanobis distance is thus unitless, scale-
| invariant, and takes into account the correlations of the
| data set._
|
| https://en.wikipedia.org/wiki/Mahalanobis_distance
| Dylan16807 wrote:
| If all your dimensions are equal to each other then it
| gives the same result as Euclidean distance? I don't think
| this counts as better, then.
| CrazyStat wrote:
| Mahalanobis distance is just a way of stretching Euclidian
| space to achieve a certain sort of isotropy (it normalizes
| an ellipsoid to the unit sphere). It is built on top of
| Euclidian distance and is not an alternative to it.
| anon_123g987 wrote:
| Euclidian distance works well in 2D and 3D as special
| cases. I would say Mahalanobis distance is its
| generalization (yes, built on top of it), which works
| better in the multidimensional (multivariate) case.
| srean wrote:
| Mahalanobis distance isn't that different from euclidean
| distance at all as far as effects of dimensions is
| concerned it just applies a stretch, rotation or more
| accurately a linear transformation to the space.
|
| In short, much that I love Mahalanobis distances' many
| properties it does zilch for dimensionality.
| CrazyStat wrote:
| No. Mahalanobis distance is not an alternative to
| Euclidian distance because it's not even measuring the
| same kind of distance. The are incommensurate, both
| figuratively and literally: Mahalanobis distance is
| unitless while Euclidian distance is not.
|
| Euclidian distance measures the distance between two
| points, while Mahalanobis measures the distance between a
| distribution (canonically multivariate normal) and a
| point. Mahalanobis distance is not a generalization if
| Euclidian distance, it's an altogether different concept
| of distance that doesn't even make sense without talking
| about a distribution with mean and covariance matrix.
| anon_123g987 wrote:
| What has more seeds, an apple or a fruit?
| CrazyStat wrote:
| What's a better fruit, an apple or an apple pie?
|
| Like Mahalanobis distance, apple pie is not a fruit and
| is not a generalization of an apple.
| srean wrote:
| I agree about your larger relevant point but the
| following that you say is bit of a red herring
|
| > Euclidean distance measures the distance between two
| points, while Mahalanobis measures the distance between a
| distribution (canonically multivariate normal) and a
| point
|
| In a discussion about metric and metric spaces we dont
| care about those things, its abstracted out and
| considered irrelevant. All that matters is that we have a
| set of 'things' and a distance between pairs of such
| things that satisfies the properties of being a distance
| (more precisely, properties of being a metric).
|
| @CrazyStat (I cannot respond to your comment so leaving
| it here)
|
| I think you overlooked
|
| > things that satisfies the properties of being a
| distance (more precisely, properties of being a metric).
|
| that I wrote. Of course it has to satisfy the properties
| of being a metric. The red herring, as far as
| dimensionality is concerned, is the complaint that
| Mahalanobis is defined over distributions while Euclidean
| is over points.
|
| The part about MD that you get absolutely right is its
| nothing but Euclidean distance in a space that has been
| transformed by a linear transformation. MD (the version
| with sqrt applied) and ED aren't that different,
| especially so in the context of dimensionality
|
| @CrazyStat response to second comment.
|
| It indeed isnt, its just Euclidean distance under linear
| transformation. I was just quoting you, you had said
|
| > while Mahalanobis measures the distance between a
| distribution
|
| My point was even it is defined for distributions its not
| really relevant.
|
| > Mahalanobis "distance" is more closely related to a
| likelihood function than to a true distance function.
|
| That's a subjective claim, and open to personal
| interpretation. Mathematically MD is indeed a metric
| (equivalently a distance) and it does show up in the log
| likelihood function. Mahalanobis was a statistician, but
| MD is a bonafide distance in any finite dimensional
| linear space, with possible extensions to infinite
| dimensional spaces by way of a positive definite kernel
| function (or equivalently, the covariance function of a
| Gaussian process)
| CrazyStat wrote:
| MD isn't defined over distributions, though. There are
| perfectly good distance metrics defined over
| distributions, but MD isn't one of them. It's a
| "distance" between one distribution and one point, not
| between two distributions or two points.
|
| Mahalanobis "distance" is more closely related to a
| likelihood function than to a true distance function.
| CrazyStat wrote:
| A function that measures the distance between two
| different classes of 'things' (distribution and point, in
| this case) is necessarily not a distance metric. It
| trivially fails to satisfy the triangle inequality,
| because at least one of d(x,y), d(x,z), d(y,z) will be
| undefined--no matter how you choose x, y, z you'll end up
| either trying to measure the distance between two points
| or the distance between two distributions, neither of
| which can be handled.
|
| This is not a red herring, it's a fundamental issue.
| amitport wrote:
| Which distance would you chose? And to what purpose?
| nealabq wrote:
| For the orthoplices, the kissing sphere in the middle pokes
| through the facets in dimension 12.
|
| I don't know when that happens with the simplices. I assume the
| middle sphere pokes thru before it does with the hypercubes,
| since the simplices are pointier.
| gjm11 wrote:
| I think that isn't the case. The simplices are pointier, which
| means that the "corner spheres" don't go so far into their
| corners.
|
| If I've done my calculations right, putting n+1 n-spheres in
| the corners of an n-simplex with unit sides so that they're
| tangent to one another gives them a radius of 1 /
| (sqrt(2n(n+1)) + 2), and then if you put a sphere in the middle
| tangent to all those it has radius [sqrt(2n/(n+1)) - 1] times
| this.
|
| So for very large n, the "corner spheres" have radius of order
| 1/n, and the "centre sphere" has radius about sqrt(2)-1 times
| the radius of the "corner spheres", and both of these -> 0 as n
| -> oo.
|
| (But! "If I've done my calculations right" is an important
| condition there. I make a lot of mistakes. If you actually care
| about the answer then you should check it.)
| woopwoop wrote:
| Vitali Milman apparently drew high-dimensional convex bodies as
| "spiky" to try to get at this intuition. So the n-dimensional
| cube in this case would look like a starfish, with the balls
| inscribed in the subdivided cubes way out in the tentacles. When
| you draw it this way, of course the middle ball is not contained
| in the cube (it is hard to reason precisely about this picture,
| it does not encode an obvious precise analogy).
| jjgreen wrote:
| It is disturbing to find one's intuition failing in higher
| dimensions.
| nabla9 wrote:
| It fails in multiple ways.
|
| For example the volume (hypervolume) gets concentrated close to
| the surface of the sphere when dimension grows. For example, if
| you have symmetric multidimensional probability distributions
| around the zero it becomes weird.
| kgwgk wrote:
| What does become weird?
| malux85 wrote:
| exciting*
| Jeff_Brown wrote:
| Yes! Let's list more!
|
| I've got a couple, maybe (depends on your intuition i guess):
|
| In 4d a (topological) sphere can be knotted.
|
| Hyugens's principle: When a wave is created in a field in
| N-dimensional space, if N is even, it will disturb an ever-
| expanding region forever (think a pebble hitting a pond's
| surface) whereas if N is odd the wavefront will propagate
| forever but leave no disturbances in its wake (think of a
| flashbulb, or of someone shouting in an infinite space full of
| air but no solids to echo off of).
| datameta wrote:
| Potentially a typo in there - one of the cases must be odd?
| Jeff_Brown wrote:
| You were right, it's fixed now, thanks!
| smegsicle wrote:
| i can see why you can't tie a circle in a knot, but why can't
| you tie a regular sphere into a knot?
| Jeff_Brown wrote:
| I should perhaps have said "a (topological) sphere can _be_
| a knot ". Corrected, thanks!
| contravariant wrote:
| > In 4d you can tie a sphere in a knot.
|
| Arguably you can do that in 3D, if you accept the horned
| sphere as a knot. Though I suppose that does raise the
| question of what you are willing to call a sphere.
|
| Regardless it's an embedding of a sphere that cannot be
| deformed into a unit sphere so I think the analogy holds.
| feoren wrote:
| Measure a group of humans on N traits and take the individual
| average of each trait. For surprisingly small N (think
| 10-ish, but obviously depending on your group size), it's
| highly likely that no human in your group (or even in
| existence) falls within 10% of the average in every trait.
| This is roughly equivalent to the statement that less and
| less of the volume of an N-sphere is near the center as N
| increases.
|
| Sometimes called "the flaw of averages". Of course I learned
| about this from another HN post recently:
|
| https://www.thestar.com/news/insight/2016/01/16/when-us-
| air-...
| jjgreen wrote:
| You want a list? https://mathoverflow.net/questions/180846/
| Jeff_Brown wrote:
| aw hells yes
| codegladiator wrote:
| https://www.youtube.com/watch?v=mceaM2_zQd8
| macilacilove wrote:
| In lower dimensional projections the inner sphere will be so big
| that it overlaps with the smaller spheres. So much so that it
| intersects with the enclosing cube too. It will not develop the
| weird spike protein things like in the illustration in any
| projection.
| f0xtrot wrote:
| I have my doubts that anything changes by adding another
| dimension, does the distance really change from 2d to 3d? I'm no
| good at activley imagining larger dimensions than that tho.
|
| None the less, it's a fun thought excercise! Thanks
| h2odragon wrote:
| I understand some of these words. Then there's another voice
| saying "this explains turbulence" but it can't explain further.
| dan_mctree wrote:
| "This means that as the dimension grows, the central sphere will
| grow in radius,"
|
| Am I missing something? It to me that only r*D grows as we
| increase the dimension, not r by itself. Since we don't seem to
| get r > 2a for any D, I don't really get the conclusion that it's
| sticking out of the cube
| phantasilide wrote:
| Does anyone have further references to pieces that examine this
| property? I read the linked paper from Strogatz which showed a
| similar geometry for the basins of coupled oscillators. Though it
| was interesting it did not provide more insight.
| TuringTest wrote:
| These are the alien geometries on other dimensions that lead to
| madness in Lovecraft's work, right?
| legohead wrote:
| Do dimensions even really exist?
| srean wrote:
| What do you mean by exist ? Do real numbers exist ?
| 0x264 wrote:
| Oh yes, they do :)
| legohead wrote:
| Don't know why I got downvoted, it was a serious question.
|
| Can you give an example of how we've made use of the 4th,
| 5th, 6th dimension, etc.?
| [deleted]
| gorloth wrote:
| Higher dimensional geometry can show up in lower
| dimensional problems. This numberphile video
| (https://www.youtube.com/watch?v=6_yU9eJ0NxA) involves a
| puzzle about throwing darts at a dart board which is solved
| by using the volumes of 4+ dimensional spheres.
| f0xtrot wrote:
| doesn't 4th just include time with a 3d object. Would
| things like waves/pulses fall under that? 5th seems to be
| used in the Kaluza-Klein theory[0] for gravitation and
| electromagnetism.
|
| [0]
| https://en.m.wikipedia.org/wiki/Kaluza%E2%80%93Klein_theory
|
| edit: I agree, I don't know why you got down voted. It was
| a thought provoking question.
| cs702 wrote:
| Yes. As the number of dimensions n increases above 3, the
| interior angles at each vertex of an n-cube get smaller and
| smaller[a], while the n-cube's hypervolume gets more and more
| concentrated near its hypersurface, which is itself a
| _hypervolume_ of n-1 dimensions.[b]
|
| Lines and areas are different animals; we cannot reason about
| them apples-to-apples; we know this intuitively. Areas and
| volumes are different animals too; we cannot reason about them
| apples-to-apples; we know this intuitively. Similarly,
| n-dimensional objects and (n+1)-dimensional objects are different
| animals; we cannot reason about them apples-to-apples.
|
| As human beings, we find it so difficult to reason "visually"
| about higher dimensional spaces, in part, I believe, because our
| puny little brains have spent a lifetime learning to model three
| dimensions (with a fourth dimension, time, flowing only in one
| direction).
|
| --
|
| [a] See this comment by scatters:
| https://news.ycombinator.com/item?id=29969181
|
| [b] See this old thread for intuitive explanations about how and
| why this happens with n-spheres as we increase the number of
| dimensions n: https://news.ycombinator.com/item?id=15676220
| denton-scratch wrote:
| Back in the 70's, Martin Gardner published an article in his
| Mathematical Games column in the SciAm, about visualising
| rotating hyperspheres and hypercubes. The way I remember it, his
| imaginary friend Dr. Morpheus (or something) had shown him colour
| animations of these rotating objects. There were a couple of
| stills in the article. The 4D objects were of course projected
| down to 2D.
|
| I've played with an animation of a wireframe hypercube that you
| could rotate around different axes. It was quite mind-boggling.
| But Gardner particularly raved about the mind-altering effect of
| viewing a rotating hypersphere. I've always wanted to view that,
| but I never heard anything about it again. It seems to me that a
| 2D projection of a hypersphere must look to all intents and
| purposes like a sphere.
|
| Does anyone know where that article might be archived? Or where I
| can view an animation of a rotating hypersphere?
| strgrd wrote:
| https://www.youtube.com/results?search_query=rotating+hypers...
| Lots of animations come up on YouTube.
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