one of our favorite obscure mathematical facts has got to be that higher-dimensional spheres are pointy
-F
long, higher-dimensional geometry
@dodec Consider first, four circles of equal size, on the corners of a square, such that the circles just touch each other. Now consider the size of the largest circle you can fit in the space in the middle, the smaller of the two circles that are tangent to all four other circles. It's pretty small.
Now consider the three-dimensional analog of that, eight spheres on the corners of a cube. The sphere that fits in the middle there (again, the smaller of the two that are tangent to all eight) is still smaller than the other spheres, but its radius is larger than that of the 2d circle.
Now consider the four-dimensional case. It works out that the inner 3-sphere (4-sphere? i always forget if a 4-dimensional sphere is a 3-sphere (for its surface) or a 4-sphere (for its volume)) is actually exactly the same size as the outer ones. So it's also tangent to every side of the hypercube.
In five dimensions, the middle 4-sphere actually doesn't even fit inside the 4-cube, it pokes out the faces! and it only gets bigger from there
-F
long, higher-dimensional geometry
I'm with you so far, but I'm not sure what that has to do with pointiness.
re: long, higher-dimensional geometry
@dodec separately (but probably relatedly in some way we can't be bothered to figure out), the "volume" of a higher-dimensional sphere shrinks extremely fast--it grows as πⁿ/n! (where n is half the dimension), and the factorial wins out here, as factorials tend to do
so you have a shape that reaches out really far while having a really small volume
-F
re: long, higher-dimensional geometry
Yeah, that makes sense. And, of course, they're still round. Higher-dimensional geometry is extremely weird.
re: long, higher-dimensional geometry
@dodec Round is spiky in 100-dimensional space!
-F
long, higher-dimensional geometry
@Felthry @dodec Actually, this is the wrong way to think about it! High-dimensional spheres aren't pointy, no part of them is any further from their centre than any other, so there's nothing you could conceive of as a "point". It's actually the *hypercube* that's pointy, as its corners get further and further from its centre relative to its faces. And this is why the higher-dimensional sphere can grow so much - the bounding spheres keep moving further away!
long, higher-dimensional geometry
@Felthry @dodec If you want a lower-dimensional visualisation for the weird shape of a high-dimensional hypercube, imagine a cube but with all the corners stretched out (or equivalently, all the faces bowing in). If you do the sphere construction on this warped cube, well, of *course* the centre sphere sticks out, how could it not? The corners are so far out!
@Felthry
That sounds fascinating. Why are they pointy?