That bit about the sphere inside the cube reminds me of something that I once heard in math class.

First, a bit of terminology: in some branches of mathematics, it's useful to make a a distinction between "sphere" and "ball". A sphere only includes the points on the surface of the object in question, while a ball includes both the surface and the inside of the sphere. So it's a ball, not a sphere, that's being dropped into the cube.

Now, you notice how a three-dimensional ball takes up less of the space inside of the cube than a two-dimensional-ball (a circle) does of a two-dimensional cube (a square)? Well, if you try the same thing with 4-dimensional objects, the 4-dimensional ball will fill even less space. In fact, if you take the limit of the size of the unit ball as the number of dimensions goes to infinity, it actually goes to zero. Therefore, infinite dimensional balls contain no space whatsoever.

So, as my math professor wisecracked, if you ever meet a man from the 23rd dimension, his balls are very small.