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Comment by unholiness | Hacker News Reader

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-...

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Related phenomenon in ML: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality

Don'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

Related: 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.

> 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!)

This is a fantastically intuitive walkthrough.

It was discussed on HN a few months ago: https://news.ycombinator.com/item?id=16523786

Hyper-spheres are a recurrent theme, see for example https://news.ycombinator.com/item?id=3995615

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