[HN Gopher] An n-ball Between n-balls
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An n-ball Between n-balls
Author : Hugsun
Score : 155 points
Date : 2024-10-09 15:50 UTC (7 hours ago)
(HTM) web link (www.arnaldur.be)
(TXT) w3m dump (www.arnaldur.be)
| steventhedev wrote:
| This is a really good demonstration of the curse of
| dimensionality[0]
|
| [0]: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality
| robwwilliams wrote:
| Impressive, helpful, and now time to rebuild my own embeddings so
| I can grasp that red n-ball with my new n-D hands.
| joaquincabezas wrote:
| wow discovering Hamming's lecture was enough for me! so good
| bt1a wrote:
| I am struggling to juggle the balls in my mind. Are there any
| stepping-stone visual pieces like this to hopefully get me there?
| Very neat write-up, but I can't wait to share the realized
| absurdity of the red ball's green box eclipsing in our 3D
| intersection of the fully diagonalized 10D construct
| pfortuny wrote:
| The hypercube is the strange thing, not the red sphere. Placing
| the blue spheres tangent to the hypercube is an artificial
| construct which only "bounds" the red sphere in small
| dimensions. Our intuition is wrong because we think of the
| problem the wrong way ("the red sphere must be bounded by the
| box", but there is no geometrical argument for that in n
| dimensions).
| drdeca wrote:
| Why did I imagine that this would be about two shapes that are
| merely _topologically_ n-balls, each having part of their
| boundary be incident with one of the two hemi(n-1)-spheres of the
| boundary of an n-ball (and otherwise not intersecting it)? (So
| like, in 3D, if you took some ball and two lumps of clay of
| different colors, and smooshed each piece of clay over half of
| the surface of the ball, with each of the two lumps of clay
| remaining topologically a 3-ball.)
|
| I don't know that there would even be anything interesting to say
| about that.
| Hugsun wrote:
| I can't tell you why you imagined that, but that's pretty funny
| nevertheless.
| mbowcut2 wrote:
| Numberphile did a video on this a while back.
| https://youtu.be/mceaM2_zQd8?si=0xcOAoF-Bn1Z8nrO
| chadhutchins10 wrote:
| Anyone else click just to slide some animations?
| Hugsun wrote:
| This guy gets it!
| Imustaskforhelp wrote:
| Can I just say how my mind is utterly blown by the animations
| Hugsun wrote:
| Thank you <3 The trigonometry involved was pretty intense at
| times.
| WhitneyLand wrote:
| Both ChatGpt 4.o and Claude failed to answer
|
| "...At what dimension would the red ball extend outside the box?"
|
| If anyone has o1-preview it'd be interesting to hear how well it
| does on this.
| V__ wrote:
| This was the prompt I gave o1-preview:
|
| > There is a geometric thought experiment that is often used to
| demonstrate the counterintuitive shape of high-dimensional
| phenomena. We start with a 4x4 square. There are four blue
| circles, with a radius of one, packed into the box. One in each
| corner. At the center of the box is a red circle. The red
| circle is as large as it can be, without overlapping the blue
| circles. When extending the construct to 3D, many things
| happen. All the circles are now spheres, the red sphere is
| larger while the blue spheres aren't, and there are eight
| spheres while there were only four circles.
|
| > There are more than one way to extend the construct into
| higher dimensions, so to make it more rigorous, we will define
| it like so: An n-dimensional version of the construct consists
| of an n-cube with a side length of 4. On the midpoint between
| each vertex and the center of the n-cube, there is an n-ball
| with a radius of one. In the center of the n-cube there is the
| largest n-ball that does not intersect any other n-ball.
|
| > At what dimension would the red ball extend outside the box?
|
| Response: "[...] Conclusion: The red ball extends outside the
| cube when n>=10n>=10."
|
| It calculated it with a step-by-step explanation. This is the
| first time I'm actually pretty stunned. It analysed the
| problem, created an outline. Pretty crazy.
| Hugsun wrote:
| I'd wager that it's in the training data.
| eniwnenahg wrote:
| Matlock, is that you?
| Hugsun wrote:
| I am not Matlock, who is that?
| lisper wrote:
| He probably meant MacGyver.
|
| https://en.m.wikipedia.org/wiki/MacGyver
| justtinker wrote:
| No, he really did mean Matlock. Grey hair guy in a white
| suit lecturing...Fictional Lawyer.
| https://www.alamy.com/stock-photo/matlock-tv-andy-
| griffith.h...
|
| Reboot on TV this year.
| lisper wrote:
| I know who Matlock is. But then it makes no sense at all.
| Sharlin wrote:
| A good way to conceptualize what's going on is not the idea that
| balls become "spiky" in high dimensions - like the article says,
| balls are always perfectly symmetrical by definition. But it's
| the _box_ becoming spiky, "caltrop-shaped", its vertices
| reaching farther and farther out from the origin as the square
| root of dimension, while the centers of its sides remain at
| exactly +-1. And the 2^N surrounding balls are also getting
| farther from the origin, while their radius remains 1/2. Now it
| should be quite easy to imagine how the center ball gets more and
| more room until it grows out of the spiky box.
| pfortuny wrote:
| Exactly: a corner of a square covers 1/4 of that part of the
| plane. A corner of a cube covers 1/8, a corner of a hypercube
| in dimension n covers just 1/(2^n) of the space. But each
| side/face/hyperface divides the plane/space/n-dim space just in
| half.
| beretguy wrote:
| Ok, now i started to understand something. Thank you.
| jheriko wrote:
| even then the edges do not suddenly curve. its just all round a
| bad analogy.
| petters wrote:
| Very good to see this as the top-voted comment. I completely
| agree that this seems like a more natural explanation of what
| is going on.
| ColinWright wrote:
| But there is another way to think of the high-dimensional balls
| where "spikey" is the right visualisation.
|
| Consider the volume of a cap. Take a plane that is 90% of the
| distance from the centre to the edge, and look at what
| percentage of the volume is "outside" that plane. When the
| dimension is high, that volume is negligible.
|
| And when the dimension is really high you can get quite close
| to the centre, and still the volume you cut off is very small.
| In our 3D world the closest thing that has this property is a
| spike. You can cut off quite close to the centre, and the
| volume excised is small.
|
| The sense in which a high-dimensional ball is _not_ a spikey
| thing is in the symmetry, and the smoothness.
|
| So when you want to develop an intuition for a high dimensional
| ball you need to think of it as simultaneously symmetrical,
| smooth, and spikey.
|
| Then think of another five impossible things, and you can have
| breakfast.
| Hugsun wrote:
| That's a good point. High dimensional objects can obtain very
| unintuitive properties, like you describe.
|
| This to me feels similar in many ways to how a corner in a
| high dimensional n-cube, although 90 degrees, no matter how
| you measure it, seems extremely spiky. As the shape does not
| increase in width, but the corners extend arbitrarily far
| away from the center. A property reserved for spiky things in
| 3D.
| dawnofdusk wrote:
| I think our geometric intuition could never be good for a
| high dimensional object. Consider a sphere in 3D. It's
| represented by the points which satisfy x^2 + y^2 + z^2 = 1.
| Because there are only three coordinates, knowing the value
| of one of them greatly reduces the possibilities for the
| other two. For example, if I know z is close to 1, then my
| point is close to the north pole (x and y are both close to
| zero).
|
| However, if I have the n-sphere x_1^2 + x_2^2 + ... + x_n^2 =
| 1, knowing the value of x_1 gives me very little information
| about all the other coordinates. And humans' interaction with
| the geometry is reality is somewhat limited to manipulating
| one coordinate at a time, i.e., our intuition is for built on
| things like moving our body linearly through space, _not_
| dilating the volume or surface area of our bodies.
| beretguy wrote:
| I... can't.
| ColinWright wrote:
| For other HN discussions of this phenomenon you can see some
| previous submissions of another article on it.
|
| That article doesn't have the nice animations, but it is from 14
| years ago ...
|
| https://news.ycombinator.com/item?id=12998899
|
| https://news.ycombinator.com/item?id=3995615
|
| And from October 29, 2010:
|
| https://news.ycombinator.com/item?id=1846682
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