An Adventure in the Nth Dimension
On the mystery of a ball that fills a box, but vanishes in the vastness of higher dimensions
The Dimensions of the Problem
The arithmetic behind all these results is straightforward; attaching meaning to the numbers is not so easy. In particular, I can see numerically—by comparing powers of π with factorials—why the unit ball’s volume reaches a maximum at n=5. But I have no geometric intuition about five-dimensional space that would explain this fact. Perhaps readers with deeper vision will be able to provide some insight.
The results on noninteger dimensions are quite otherworldly. The notion of fractional dimensions is familiar enough, but it is generally applied to objects, not to spaces. For example, the Sierpinsky triangle, with its endlessly nested holes within holes, is assigned a dimension of 1.585, but the triangle is still drawn on a plane of dimension 2. What would it mean to construct a space with 5.2569464 mutually perpendicular coordinate axes? I can’t imagine—and that’s not just a figure of speech.
Another troubling question is whether it really makes sense to compare volumes across dimensions. Each dimension requires its own units of measure, and so the relative magnitudes of the numbers attached to those units don’t mean much. Is a disk of area 10 square centimeters larger or smaller than a ball of volume 5 cubic centimeters? We can’t answer; it’s like comparing apples and orange juice.
Nevertheless, I believe there is indeed a valid basis for making comparisons. In each dimension volume is to be measured in terms of a standard volume in that dimension. The obvious standard is the unit cube (sometimes called the “measure polytope”), which has a volume of 1 in all dimensions. Starting at n=1, the unit ball is larger than the unit cube, and the ball-to-cube ratio gets still larger through n=5; then the trend reverses, and eventually the ball is much smaller than the unit cube. This changing ratio of ball volume to cube volume is the phenomenon to be explained.