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Comment by ithinkso | Hacker News Reader

The way I think about higher dimensions is just by looking what a sphere of radius r looks like in cartesian coordinates

x^2 + y^2 = r^2 in 2D

x^2 + y^2 + z^2 = r^2 in 3D

x^2 + y^2 + z^2 + t^2 = r^2 in 4D

If that leads to some weird behaviors (spheres are very 'spike-y') then so be it, I don't understand why intuition from 3D is important

Things gets 'weirder' in higher dim manifolds but not really, it's only hard if you want to 'see' it in 3d Euclidean

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It's often repeated but describing spheres as "spikey" isn't right at all. The argument made is that they seem spikey in rectangular coordinates. But that's actually a statement about cubes, which are spikey. That's easy to see even just considering 2 and 3 dimensions : the corner of a cube is better for stabbing than a side. But the sphere is rotationally symmetric in all dimensions and is not spikey in the least bit.

What do you mean by "spike-y"? That's not how I think of higher dimensional spheres at all.

Dimension 2, put 4 circles radius 1 at (1, 1), (-1, 1), (1, -1), (-1, -1). In the centre, put the largest circle you can. Then from the outside, you can't "touch" the inner circle.

Dimension 3, put 8 spheres radius 1 at (1, 1, 1), ... Then from the outside you can touch the inner sphere (and it's a bit bigger).

Once you get to a certain dimension (10 IIRC), the inner sphere is no longer in the convex hull of the outer spheres, it is "poking out" of the arrangement.

Spiky like that.

Oh they are very spike-y, well, my point in the above post is to just solve the eq but easier 'visualization' would be [0]

By the way, this is a similar phenomena to the 'curse of dimensionality' [1]

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

[1] https://en.wikipedia.org/wiki/Curse_of_dimensionality

Most of the volume is near the edge of the sphere in higher dimensions. Closer to soap bubbles than what we consider to be true spheres.

How does that help you?

It helps me in the sense that if some object is moving (I can artificially make it move for the sake of the argument) then I just change it's coordinates instead o how it 'would look like to m eyes', I don't know, that makes me sleep easier

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