one of our favorite obscure mathematical facts has got to be that higher-dimensional spheres are pointy
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@Felthry

That sounds fascinating. Why are they pointy?

@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
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@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
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