Spheres in higher dimensions are kind of “spiky” compared to lower dimensions. It kind of makes sense, if you think a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = 1 for a 7d sphere. Well if a^2 is 0.9, that means the remaining 0.1 has to be shared across multiple other dimensions instead of just 2 for a 3d sphere. So there’s less bulging in a way.

Here are two points on a 3d sphere: (1, 0, 0), (1/sqrt(3), 1/sqrt(3), 1/sqrt(3))

Here are three points on a 7d sphere: (1, 0, 0, 0, 0, 0, 0), (1/sqrt(3), 1/sqrt(3), 1/sqrt(3), 0, 0, 0, 0), (1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7), 1/sqrt(7)).

You can kind of see how this last point “bulges” less? Anyway, that’s kind of my intuition.

Because you’re comparing apples and oranges. The issue is that we are using the same word, “volume” in a more generalized way, but in actuality the “volumes” of different dimensional spheres are measures of different things. It doesn’t make sense to say that they are smaller or larger than each other.

The easiest way to see this is with the “volume” of a 2-dimensional sphere (also known as a “circle”) and that of a 3-dimensional sphere (also known as a “sphere”).

Yes, if you calculate the “volume” of each, the magnitude of the sphere’s volume is smaller than that of the circle’s (given the same radius) but one isn’t “smaller” than the other because they are different units.

The circle’s volume, which we would more commonly call its area, is measured in square units whereas the sphere’s volume is measured in cube units. Think about this with real units: is 2 square meters smaller or larger than 1 cubic meter?

It’s a borderline nonsensical question because they are measuring different things: area vs volume.

In this same vein, the volumes of different dimensional spheres are measuring different things. We just say “volume” collectively for ease of description when we are talking about the concepts in general. For the same reason we just call them all “spheres” instead of inventing a new name for each higher dimensional shape.

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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