3 ms·
High dimensional spheres are very counterintuitive! The volume of a n-dimensional unit sphere goes to zero as n increases, almost all the volume is contained ne
by unholiness 8y ago
High dimensional spheres are very counterintuitive! The volume of a n-dimensional unit sphere goes to zero as n increases, almost all the volume is contained near the surface.
Some interesting musings in this realm:
https://marckhoury.github.io/counterintuitive-properties-of-high-dimensional-space/ https://marckhoury.github.io/counterintuitive-properties-of-...
- alexpetralia 8y agoThis is a fantastically intuitive walkthrough.
- kgwgk 8y agoIt was discussed on HN a few months ago: https://news.ycombinator.com/item?id=16523786 https://news.ycombinator.com/item?id=16523786 Hyper-spheres are a recurrent theme, see for example https://news.ycombinator.com/item?id=3995615 https://news.ycombinator.com/item?id=3995615
- Lxr 8y agoRelated phenomenon in ML: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality https://en.m.wikipedia.org/wiki/Curse_of_dimensionality
- jedimastert 8y agoDon't forget 3Blue1Brown's incredible video about visualizing higher dimensions[1]. He's getting to xkcd levels of "Of course there's a relevant video" [1]: https://www.youtube.com/watch?v=zwAD6dRSVyI https://www.youtube.com/watch?v=zwAD6dRSVyI
- TorKlingberg 8y agoRelated: In 4D and higher space, the concept of rotating around an axis does not make sense. In 3D there are three ways to turn: yaw, pitch and roll. In 4D, there are not 4 but 6 ways.
- deleted 8y ago[deleted]
- JadeNB 8y ago> Related: In 4D and higher space, the concept of rotating around an axis does not make sense. I think that this is like saying "rotating in 3D doesn't make sense"; it's not so much that it doesn't make sense as that it's not uniquely specified. (Then there's the fact, which maybe is what you meant (since you referenced the dimension of SO(4)), that a 4D Euclidean rotation need not fix an axis at all!)