An Adventure in the Nth Dimension
On the mystery of a ball that fills a box, but vanishes in the vastness of higher dimensions
I had thought that I understood Bellman’s curse: Both the n-ball and the n-cube grow along with n, but the cube expands faster. In fact, the curse is far more damning: At the same time the cube inflates exponentially, the ball shrinks to insignificance. In a space of 100 dimensions, the fraction of the cubic volume filled by the ball has declined to 1.8×10–70. This is far smaller than the volume of an atom in relation to the volume of the Earth. The ball in the box has all but vanished. If you were to select a trillion points at random from the interior of the cube, you’d have almost no chance of landing on even one point that is also inside the ball.
What makes this disappearing act so extraordinary is that the ball in question is still the largest one that could possibly be stuffed into the cube. We are not talking about a pea rattling around loose inside a refrigerator carton. The ball’s diameter is still equal to the side length of the cube. The surface of the ball touches every face of the cube. (A face of an n-cube is an (n–1)-cube.) The fit is snug; if the ball were made even a smidgen larger, it would bulge out of the cube on all sides. Nevertheless, in terms of volume measure, the ball is nearly crushed out of existence, like a black hole collapsing under its own mass.
How can we make sense of this seeming paradox? One way of understanding it is to acknowledge that the ball fills the middle of the cube, but the cube doesn’t have much of a middle; almost all of its volume is away from the center, huddling in the corners. A simple counting argument gives a clue to what’s going on. As noted above, the ball touches the enclosing cube at the center of each face, but it does not reach out into the corners. A 100-cube has just 200 faces, but it has 2100 corners.
Another approach to understanding the collapse of the n-ball is to imagine poking skewers through the cube along various diameters. (A diameter is any straight line that passes through the center point.) The shortest diameters run from the center of a face to the center of the opposite face. For the cube enclosing a unit ball, the length of this shortest diameter is 2, which is both the side length of the cube and the diameter of the ball. Thus a skewer on the shortest diameter lies inside the ball throughout its length.
The longest diameters of the cube extend from a corner through the center point to the opposite corner. For an n-cube with side length s=2, the length of this diameter is 2 times the square root of n. Thus in the 100-cube surrounding a unit ball, the longest diameter has length 20; only 10 percent of this length lies within the ball. Moreover, there are just 100 of the shortest diameters, but there are 299 of the longest ones.
Here is still another mind-bending trick with balls and boxes to suggest just how weird space becomes in higher dimensions. I learned of it from Barry Cipra, who published a description in Volume 1 of What’s Happening in the Mathematical Sciences (1991). On the plane, a square with sides of length 4 will accommodate four unit disks in a two-by-two array, with room for a smaller disk in the middle; the radius of that smaller disk is √2 – 1. In three dimensions the equivalent 3-cube fits eight unit balls, plus a smaller ninth ball in the middle, whose radius is √3 – 1. In the general case of n dimensions, the box has room for 2n unit n-balls in a rectilinear array, with one additional ball in the vacant central space, and the central ball has a radius of √n – 1. Look what happens when n reaches 9. The “smaller” central ball now has a radius of 2, which makes it twice the size of the 512 surrounding balls. Furthermore, the central ball has expanded to reach the sides of the bounding box, and will burst through the walls with any further increase in dimension.