The volume of spheres of the same diameter approaches zero as dimensionality approaches infinity, I intuitively don’t understand why.
In: Mathematics
The “cubes” of side length 1 always have a “volume” of 1, right? So think about inscribing hyperspheres in these hypercubes. Each dimension you add is another portion of the whole cube that you are carving away to be left with a sphere.
Or think about it in 2 and 3 dimensions first. You have a circle inscribed in a square. The area within the circle is pi/4.
Now bump it up to 3 dimensions. If you just “stretch” it into the third, you just have a cylinder inscribed in a cube, not a sphere. And the volume of that cylinder is in the same ratio, pi/4. To get a sphere, you have to carve away more of it, because the distance to the top and bottom edges is cube root of x^3 + y^3 + z^3, not just x^2 + y^2. And the remaining volume ends up being smaller, in this case pi/6.
Adding each additional dimension does the same thing, it’s just harder (impossible) to visualize. You are “carving away” more each time.
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