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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

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

AnimalMuppet 5y ago
What do you mean by "spike-y"? That's not how I think of higher dimensional spheres at all.
- ithinkso 5y ago
  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 https://www.youtube.com/watch?v=mceaM2_zQd8 [1] https://en.wikipedia.org/wiki/Curse_of_dimensionality https://en.wikipedia.org/wiki/Curse_of_dimensionality
- 3pt14159 5y ago
  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.
  - lupire 5y ago
    how is that spikey? how is a soap bubble not a sphere? you mean a pile of spheres od differnt sizes? a hypersphere is a smooth stack of spheres, just as a sphere is a smooth pile of circles.
- jjgreen 5y ago
  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.
sorokod 5y ago
How does that help you?
- ithinkso 5y ago
  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
tgb 5y ago
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.

AnimalMuppet 5y ago

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

ithinkso 5y ago
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 https://www.youtube.com/watch?v=mceaM2_zQd8 [1] https://en.wikipedia.org/wiki/Curse_of_dimensionality https://en.wikipedia.org/wiki/Curse_of_dimensionality
3pt14159 5y ago
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.
- lupire 5y ago
  how is that spikey? how is a soap bubble not a sphere? you mean a pile of spheres od differnt sizes? a hypersphere is a smooth stack of spheres, just as a sphere is a smooth pile of circles.
jjgreen 5y ago
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.

sorokod 5y ago

How does that help you?

ithinkso 5y ago
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

tgb 5y ago

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.