3 ms·

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

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.