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 2

^{n}. 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