This is a really good demonstration of the curse of dimensionality[0]
[0]: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality
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.
Impressive, helpful, and now time to rebuild my own embeddings so I can grasp that red n-ball with my new n-D hands.
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:
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
Can I just say how my mind is utterly blown by the animations
Anyone else click just to slide some animations?
Numberphile did a video on this a while back. https://youtu.be/mceaM2_zQd8?si=0xcOAoF-Bn1Z8nrO
wow discovering Hamming’s lecture was enough for me! so good
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.
I never understood the need to distinguish between "ball" and "sphere" in maths. Sure, one is solid and the other hollow, but why is that fact so important that you need to use a completely different word? As I understand it, you could replace every instance of "ball" in this article with "sphere" and it would still be correct.
We don't have special words for the voluminous versions of other 3D shapes, so why do spheres need one?
I… can’t.
Matlock, is that you?
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.