4 ms·

Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8 https://youtu.be/mceaM2_zQd8

by dvh 5y ago

rwmj 5y ago
It has to be said that Matt's explanation in that video is wrong. N-spheres are just spheres, it's the cubes which get "spiky", like sea-urchins. (It's a rare case where reading the Youtube comments is a good idea.)
- ColinWright 5y ago
  I agree that high dimensional cubes are "spikey", but consider this. If you stand on a high-dimensional sphere, then slice off the "cap" where you are standing, then in high dimensions the resulting cap has virtually no volume, unless your cut is a long way towards the centre of the sphere. In other words, the piece you are standing on has a very, very small volume, unless you cut off a lot of it. That's the same intuition as if you are standing on a spike. Cutting off the spike results in a solid with nearly no volume. So thinking of a sphere as being "spikey" with a spike at every location does give a better intuition for some things. But not everything ... high-dimensional stuff is just generally weird.
  - dvh 5y ago
    The problem with slicing is that here in 3d we would slice with 2d plane, but when you are in 4d, should you slice with 2d plane or 3d something? (3-1=2 so 4-1=3) or is slicing always 2d plane, says who?
    - ColinWright 5y ago
      To "slice" something you need a "plane" in one dimension lower. To get some sense of why, think about a ball in three dimensions. You can "slice off" some of its volume with a plane (one dimension less), but if you have a line all you do is have a line through it ... the ball is still connected, and you haven't sliced anything from it.
ColinWright 5y ago
I wrote a thing about this and have submitted it in the past. Actually, I've realised I submitted it more than once. Each time there was a small amount of conversation that discussed this point. If you You might be interested. https://news.ycombinator.com/item?id=12998899 https://news.ycombinator.com/item?id=12998899 : 45 comments https://news.ycombinator.com/item?id=3995615 https://news.ycombinator.com/item?id=3995615 : 59 comments I think this pre-dates Matt talking about this fun, and he may have got it from me, but he may also have got it from the same place I got it, which was (probably) Martin Gardner. https://en.wikipedia.org/wiki/Martin_Gardner https://en.wikipedia.org/wiki/Martin_Gardner

Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8 https://youtu.be/mceaM2_zQd8

by dvh 5y ago

Higher dimensional spheres are weird, especially 10+ dimensional: https://youtu.be/mceaM2_zQd8 https://youtu.be/mceaM2_zQd8

rwmj 5y ago

It has to be said that Matt's explanation in that video is wrong. N-spheres are just spheres, it's the cubes which get "spiky", like sea-urchins. (It's a rare case where reading the Youtube comments is a good idea.)

ColinWright 5y ago
I agree that high dimensional cubes are "spikey", but consider this. If you stand on a high-dimensional sphere, then slice off the "cap" where you are standing, then in high dimensions the resulting cap has virtually no volume, unless your cut is a long way towards the centre of the sphere. In other words, the piece you are standing on has a very, very small volume, unless you cut off a lot of it. That's the same intuition as if you are standing on a spike. Cutting off the spike results in a solid with nearly no volume. So thinking of a sphere as being "spikey" with a spike at every location does give a better intuition for some things. But not everything ... high-dimensional stuff is just generally weird.
- dvh 5y ago
  The problem with slicing is that here in 3d we would slice with 2d plane, but when you are in 4d, should you slice with 2d plane or 3d something? (3-1=2 so 4-1=3) or is slicing always 2d plane, says who?
  - ColinWright 5y ago
    To "slice" something you need a "plane" in one dimension lower. To get some sense of why, think about a ball in three dimensions. You can "slice off" some of its volume with a plane (one dimension less), but if you have a line all you do is have a line through it ... the ball is still connected, and you haven't sliced anything from it.

ColinWright 5y ago

I wrote a thing about this and have submitted it in the past. Actually, I've realised I submitted it more than once. Each time there was a small amount of conversation that discussed this point. If you You might be interested. https://news.ycombinator.com/item?id=12998899 https://news.ycombinator.com/item?id=12998899 : 45 comments https://news.ycombinator.com/item?id=3995615 https://news.ycombinator.com/item?id=3995615 : 59 comments I think this pre-dates Matt talking about this fun, and he may have got it from me, but he may also have got it from the same place I got it, which was (probably) Martin Gardner. https://en.wikipedia.org/wiki/Martin_Gardner https://en.wikipedia.org/wiki/Martin_Gardner