13 ms·

A high-dimensional sphere spilling out of a high-dimensional cube

299 points by EvgeniyZh 5y ago

f0xtrot 5y ago
could this be used in > light speed travel?
- f0xtrot 5y ago
  only if your travling on the surface in the 10th dimension.
jjgreen 5y ago
It is disturbing to find one's intuition failing in higher dimensions.
- malux85 5y ago
  exciting*
- Jeff_Brown 5y ago
  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).
  - smegsicle 5y ago
    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 5y ago
      I should perhaps have said "a (topological) sphere can be a knot". Corrected, thanks!
  - jjgreen 5y ago
    You want a list? https://mathoverflow.net/questions/180846/ https://mathoverflow.net/questions/180846/
    - Jeff_Brown 5y ago
      aw hells yes
  - datameta 5y ago
    Potentially a typo in there - one of the cases must be odd?
    - Jeff_Brown 5y ago
      You were right, it's fixed now, thanks!
  - contravariant 5y ago
    > 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 5y ago
    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-force-discovered-the-flaw-of-averages.html https://www.thestar.com/news/insight/2016/01/16/when-us-air-...
- nabla9 5y ago
  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 5y ago
    What does become weird?
    - nabla9 5y ago
      Consider normal distribution. n-dimensional vectors with independent N(0,1) with expectation 0. When n increases the mass of the probabilities are around a sphere of radius n^0.5, nowhere near the origin.
      - kgwgk 5y ago
        True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…
h2odragon 5y ago
I understand some of these words. Then there's another voice saying "this explains turbulence" but it can't explain further.
spenczar5 5y ago
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.
- jeeceebees 5y ago
  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 5y ago
    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 5y ago
      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 5y ago
        Aha! Right, this is pretty convincing to me. Thanks!
- jjgreen 5y ago
  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) ...
hprotagonist 5y ago
relatedly, euclidean distance is a shitty metric in high dimension.
- AnimalMuppet 5y ago
  How so? And, what is a better metric, and why?
  - hprotagonist 5y ago
    cosine similarity’s a lot better. https://stats.stackexchange.com/questions/99171/why-is-euclidean-distance-not-a-good-metric-in-high-dimensions https://stats.stackexchange.com/questions/99171/why-is-eucli...
    - CrazyStat 5y ago
      Cosine similarity is not a distance metric.
      - contravariant 5y ago
        Well it is on the unit sphere, but then it's equivalent to the euclidean metric...
        srean 5y ago
        Do I spot a geometer here ? You are indeed right but its not something that is well known.
  - anon_123g987 5y ago
    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 https://en.wikipedia.org/wiki/Mahalanobis_distance
scatters 5y ago
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.).
- shenberg 5y ago
  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 ε 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.
  - kgwgk 5y ago
    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 5y ago
    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 5y ago
      >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.
fghorow 5y ago
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 5y ago
  Symmetry.
- nealabq 5y ago
  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 5y ago
  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?
- fghorow 5y ago
  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 https://www.math.wustl.edu/~feres/highdim
- cantagi 5y ago
  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.
legohead 5y ago
Do dimensions even really exist?
- 0x264 5y ago
  Oh yes, they do :)
  - legohead 5y ago
    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.?
    - gorloth 5y ago
      Higher dimensional geometry can show up in lower dimensional problems. This numberphile video (https://www.youtube.com/watch?v=6_yU9eJ0NxA 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.
    - deleted 5y ago
      [deleted]
    - f0xtrot 5y ago
      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 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.
- srean 5y ago
  What do you mean by exist ? Do real numbers exist ?
  - legohead 5y ago
rwmj 5y ago
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 5y ago
  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 5y ago
    Doesn't Alicia Boole Stott's ability indicate that solid intuitions are plausible though? https://www.askaboutireland.ie/reading-room/life-society/science-technology/irish-scientists/alicia-boole-stott-(1860-/ https://www.askaboutireland.ie/reading-room/life-society/sci...
    - feoren 5y ago
      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?"
codegladiator 5y ago
https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8
denton-scratch 5y ago
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 5y ago
  https://www.youtube.com/results?search_query=rotating+hypersphere https://www.youtube.com/results?search_query=rotating+hypers... Lots of animations come up on YouTube.
cs702 5y ago
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 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 https://news.ycombinator.com/item?id=15676220
- zarzavat 5y ago
  It’s not just our brains that are evolutionarily unsuited to the task, but almost all of us except the few mathematicians who study these higher dimensional spaces, have basically no experience or exposure to higher dimensional spaces. If you spend a year playing 4D Minecraft (I assume without googling that this exists), I suspect your intuition would be very much improved.
emsign 5y ago
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 5y ago
  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?
nealabq 5y ago
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 5y ago
  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.)
TuringTest 5y ago
These are the alien geometries on other dimensions that lead to madness in Lovecraft's work, right?
ColinWright 5y ago
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=12998899 https://news.ycombinator.com/item?id=3995615 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.
- it_does_follow 5y ago
  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 https://youtu.be/uU_Q2a0S0zI?t=1716 1. https://press.stripe.com/the-art-of-doing-science-and-engineering https://press.stripe.com/the-art-of-doing-science-and-engine...
- capableweb 5y ago
  > 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.
macilacilove 5y ago
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.
phantasilide 5y ago
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.
- jemfinch 5y ago
  I like this video on the topic: https://youtu.be/zwAD6dRSVyI https://youtu.be/zwAD6dRSVyI
f0xtrot 5y ago
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
- f0xtrot 5y ago
  nvm - this doesnt look like it cares about distance from center points at all (what I cared about) and instead care's whether the surface area spills from the contained square 4a square at any dimension, opps.
  - mbfg 5y ago
    Watch this: https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8
growt 5y ago
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.
woopwoop 5y ago
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).
dan_mctree 5y ago
"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
juloo 5y ago
Numberphile has a video on exactly this: https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8 I love this channel :)
ssener2001 5y ago
I understand better this way. Consider a large circle in the middle in 2-D that exceeds the size of the square . The circles on all four sides will be smaller. In higher dimensions(>9), it means it has to be a similar shape.
mariusor 5y ago
I think this would be something that can be experimented with using Marc Ten Bosch's 4D Toys[1], I'm surprised nobody mentioned it. [1] https://4dtoys.com/ https://4dtoys.com/

299 points by EvgeniyZh 5y ago

f0xtrot 5y ago

could this be used in > light speed travel?

f0xtrot 5y ago
only if your travling on the surface in the 10th dimension.

jjgreen 5y ago

It is disturbing to find one's intuition failing in higher dimensions.

malux85 5y ago
exciting*
Jeff_Brown 5y ago
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).
- smegsicle 5y ago
  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 5y ago
    I should perhaps have said "a (topological) sphere can be a knot". Corrected, thanks!
- jjgreen 5y ago
  You want a list? https://mathoverflow.net/questions/180846/ https://mathoverflow.net/questions/180846/
  - Jeff_Brown 5y ago
    aw hells yes
- datameta 5y ago
  Potentially a typo in there - one of the cases must be odd?
  - Jeff_Brown 5y ago
    You were right, it's fixed now, thanks!
- contravariant 5y ago
  > 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 5y ago
  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-force-discovered-the-flaw-of-averages.html https://www.thestar.com/news/insight/2016/01/16/when-us-air-...
nabla9 5y ago
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 5y ago
  What does become weird?
  - nabla9 5y ago
    Consider normal distribution. n-dimensional vectors with independent N(0,1) with expectation 0. When n increases the mass of the probabilities are around a sphere of radius n^0.5, nowhere near the origin.
    - kgwgk 5y ago
      True, a huge volume with low density will contain more mass than a small volume with large density. But we can say that it gets “concentrated” far from the origin only in the same sense that Japan’s population is “concentrated” outside of Tokio…

h2odragon 5y ago

I understand some of these words. Then there's another voice saying "this explains turbulence" but it can't explain further.

spenczar5 5y ago

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.

jeeceebees 5y ago
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 5y ago
  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 5y ago
    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 5y ago
      Aha! Right, this is pretty convincing to me. Thanks!
jjgreen 5y ago
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) ...

hprotagonist 5y ago

relatedly, euclidean distance is a shitty metric in high dimension.

AnimalMuppet 5y ago
How so? And, what is a better metric, and why?
- hprotagonist 5y ago
  cosine similarity’s a lot better. https://stats.stackexchange.com/questions/99171/why-is-euclidean-distance-not-a-good-metric-in-high-dimensions https://stats.stackexchange.com/questions/99171/why-is-eucli...
  - CrazyStat 5y ago
    Cosine similarity is not a distance metric.
    - contravariant 5y ago
      Well it is on the unit sphere, but then it's equivalent to the euclidean metric...
      - srean 5y ago
        Do I spot a geometer here ? You are indeed right but its not something that is well known.
- anon_123g987 5y ago
  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 https://en.wikipedia.org/wiki/Mahalanobis_distance

scatters 5y ago

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.).

shenberg 5y ago
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 ε 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.
- kgwgk 5y ago
  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 5y ago
  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 5y ago
    >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.

fghorow 5y ago

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 5y ago
Symmetry.
nealabq 5y ago
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 5y ago
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?
fghorow 5y ago
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 https://www.math.wustl.edu/~feres/highdim
cantagi 5y ago
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.

legohead 5y ago

Do dimensions even really exist?

0x264 5y ago
Oh yes, they do :)
- legohead 5y ago
  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.?
  - gorloth 5y ago
    Higher dimensional geometry can show up in lower dimensional problems. This numberphile video (https://www.youtube.com/watch?v=6_yU9eJ0NxA 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.
  - deleted 5y ago
    [deleted]
  - f0xtrot 5y ago
    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 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.
srean 5y ago
What do you mean by exist ? Do real numbers exist ?
- legohead 5y ago

rwmj 5y ago

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 5y ago
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 5y ago
  Doesn't Alicia Boole Stott's ability indicate that solid intuitions are plausible though? https://www.askaboutireland.ie/reading-room/life-society/science-technology/irish-scientists/alicia-boole-stott-(1860-/ https://www.askaboutireland.ie/reading-room/life-society/sci...
  - feoren 5y ago
    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?"

codegladiator 5y ago

https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8

denton-scratch 5y ago

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 5y ago
https://www.youtube.com/results?search_query=rotating+hypersphere https://www.youtube.com/results?search_query=rotating+hypers... Lots of animations come up on YouTube.

cs702 5y ago

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 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 https://news.ycombinator.com/item?id=15676220

zarzavat 5y ago
It’s not just our brains that are evolutionarily unsuited to the task, but almost all of us except the few mathematicians who study these higher dimensional spaces, have basically no experience or exposure to higher dimensional spaces. If you spend a year playing 4D Minecraft (I assume without googling that this exists), I suspect your intuition would be very much improved.

emsign 5y ago

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 5y ago
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?

nealabq 5y ago

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 5y ago
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.)

TuringTest 5y ago

These are the alien geometries on other dimensions that lead to madness in Lovecraft's work, right?

ColinWright 5y ago

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=12998899 https://news.ycombinator.com/item?id=3995615 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.

it_does_follow 5y ago
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 https://youtu.be/uU_Q2a0S0zI?t=1716 1. https://press.stripe.com/the-art-of-doing-science-and-engineering https://press.stripe.com/the-art-of-doing-science-and-engine...
capableweb 5y ago
> 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.

macilacilove 5y ago

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.

phantasilide 5y ago

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.

jemfinch 5y ago
I like this video on the topic: https://youtu.be/zwAD6dRSVyI https://youtu.be/zwAD6dRSVyI

f0xtrot 5y ago

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

f0xtrot 5y ago
nvm - this doesnt look like it cares about distance from center points at all (what I cared about) and instead care's whether the surface area spills from the contained square 4a square at any dimension, opps.
- mbfg 5y ago
  Watch this: https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8

growt 5y ago

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.

woopwoop 5y ago

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).

dan_mctree 5y ago

"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

juloo 5y ago

Numberphile has a video on exactly this: https://www.youtube.com/watch?v=mceaM2_zQd8 https://www.youtube.com/watch?v=mceaM2_zQd8 I love this channel :)

ssener2001 5y ago

I understand better this way. Consider a large circle in the middle in 2-D that exceeds the size of the square . The circles on all four sides will be smaller. In higher dimensions(>9), it means it has to be a similar shape.

mariusor 5y ago

I think this would be something that can be experimented with using Marc Ten Bosch's 4D Toys[1], I'm surprised nobody mentioned it. [1] https://4dtoys.com/ https://4dtoys.com/