Cubes viz. Spheres

[PatrickPowers]

One often sees mentioned that the volume of the unit sphere goes to zero as the number of dimensions n increases. I felt this was more of an artifact of the measuring system than anything else. Instead let us compare spheres and cubes of equal volume. In particular let's have such a cube and sphere with a common center. We know right away that one is not the subset of the other. If they aren't transparent then what one sees is a sphere with the corners of the cube protruding. Some parts of the cube are exposed, some hidden. When n is large (greater than 30?) the diameter of the cube -- the maximal diagonal -- is always just a bit more than two times the diameter of the sphere. With a bit more work we will be able to tell very easily whether many key points are hidden or exposed. I call these key points the corners of the unit cube's subcubes.

It's convenient to center all our cubes and spheres at the origin. The corners of our unit cube are all exposed, protruding from the sphere, so all the coordinates with values of only 1/2 and -1/2 are exposed. The center is hidden so the point with all coordinates equal to 0 is hidden. The question now is, what about all the coordinates where the values may be any mixture of -1/2, 0, or 1/2? It turns out for large n there is a very simple rule. Consider the set [t,u, v, w, 0, 0, 0] with -1/2<=t,u,v,w<=1/2. We say this defines a subcube of our preferred seven dimensional unit cube with dimension four. (By symmetry its center is at the origin.) If and only if the dimension of the subcube is greater than n/4 then the corners of that subcube protrude from the sphere.

To get the geometric idea of these nested subcubes let's look at familiar 3D, even though in this case n is too small for us. A subcube of dimension two would be any solid square though the origin that is parallel or perpendicular to all sides. A subcube of dimension one would be any line though the origin that is parallel or perpendicular to all sides. (This example is murky in higher dimensions as the notion of perpendicularity is not commonly defined. Stick with the set form.)

To demonstrate this n/4 rule, start with the unit cube centered at the origin. Its volume is one for all number of dimensions n. The maximum diagonal is n^1/2.

For the sphere of the same volume and center we use Stirling's approximation for n large combined with Wikipedia formulas* to find the diameter is (2n/(pi*e))^1/2.

We want the ratio of these two quantities for n large. The n^1/2 cancel so we get a close approx. of

(max. diagonal of cube)/(diameter of sphere) = (pi*e/2)^1/2 = 2.066.

The diagonal from the center to a corner is a bit more than twice the radius of the sphere. This diagonal has the length n^1/2/2. Using our newly computed ratio, we know that any point within n^1/2/4 of the center is inside the sphere. If more than 1/4 of those restricted coordinates of -1/2, 0, and 1/2 are nonzero then the point is outside the sphere. Conversely, if 3/4 or more are zero then the point is inside the sphere.

What does that mean geometrically? We can tell whether a subcube is entirely inside the sphere as opposed to its corners protruding from the sphere.

So... for large n, all subcubes with dimensions less than or equal to n/4 are entirely inside the sphere, while subcubes with dimensions greater than n/4 have all their corners outside of the sphere.

As n grows the number of corners protruding from the sphere is 2ⁿ. Does the proportion of the cube that is outside the sphere grow with n? Is there a limit to that proportion, and if so what is it?

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*https://en.wikipedia.org/wiki/Volume_of_an_n-ball

[quickfur]

There is no limit. Even if you keep the volumes of the n-sphere and n-cube constant, as n grows without bound the proportion of the volume of the n-sphere outside the n-cube grows arbitrarily close to 1, as does the proportion of the volume of the n-cube that lies outside the n-sphere. Almost all of the volume of the n-cube will lie close to its vertices, which as n grows are unboundedly far from the origin, such that proportionally the volume in its interior around its center is almost zero. Conversely, the proportion of the n-sphere's volume around its center will also be arbitrarily small, most of its volume being concentrated near its surface.

Another way to think about this is to consider an n-cube that's inscribed inside an n-sphere. Obviously, the vertices of this inscribed n-cube will be of the form <±x, ±x, ±x,...> for some x>0, such that its Euclidean norm is equal to the radius of the n-sphere. If we keep the radius constant, then as n increases without bound x will decrease without bound (the more components the vector has, the smaller n·xⁿ must be in order to remain equal to the square of the radius). Meaning that as the number of dimensions increase, the volume of this n-cube will decrease towards 0. So the higher the dimension, the more volume of the n-sphere will lie away from an n-cube's corners, whereas the more the n-cube's volume will be concentrated around its corners. Therefore, the intersection between the two will decrease toward 0.

In the limiting case, the intersection Z of the ∞-sphere with the line p = <1,1,1,..>*n will have infinitesimal coordinates, i.e., its coordinates are infinitesimally small yet its radius is non-zero. In the same direction, this line will intersect with a unit ∞-cube at <1,1,1,...>, which will be infinitely distant from Z. Since practically all of the volume of the ∞-cube lies near its corners, this means practically all of its volume lies outside the ∞-sphere.

[PatrickPowers]

As can be seen here*, it is considered a mystery that the constants in formulae for the volume of n-spheres have a maximum at n=5. I say that the maximum with n=5 is an artifact of an arbitrary feature. To see this all you have to do is use the diameter of the spheres instead of the radius. d/2 = r. Then the constants that appear in the volume formulas decrease monotonically.

[sqrt(pi)/2]^n/n!

*https://mathoverflow.net/questions/53119/volumes-of-n-balls-what-is-so-special-about-n-5

Geometrically what is going on is that for n<5 the unit cube lies entirely inside the unit sphere. At n=5 the diagonal of the sphere is long enough for the corners to start poking out, but not enough to stop the increase. After that it's all over.

By choosing the appropriate "unit sphere" you can get a maximum for any n you like.