My research involves a field called dimensionality reduction. As such, I do a lot of thinking about dimensions and, unfortunately, have to think about dimensions greater than three. If you have never thought of a cube in 4 dimensions, try it now. Go ahead. I’ll wait.
That wasn’t much fun, was it? You weren’t very successful, were you? It’s ok. I never get far either.
Today I want to bring a crazy quirk about dimensions to your attention. We are going to look at the volume of the n-sphere, which is just a fancy term for an ordinary sphere in different dimensions (n+1 dimensions, to be precise). Now, the volume of the (n-1)-sphere is given by a messy equation,
Rather than explain the gamma function, I can use the helpful table given below to explain my quirk (thanks, Wikipedia!). Here are the volumes for a sphere of radius R in dimensions 0 through 9. (Note that, when we say “volume,” we refer to how much “stuff” we can fit inside a sphere in a given dimension. For example, the volume of a line segment is just its length.)
Notice that the volume scales with the radius raised to the power of the dimension? Notice how the constant that multiplies the radius peaks between 5 and 6 (actually, it peaks at n = 5.2569464) dimensions? Does this seem weird to you?
Let me explain in words. This says that if you take a sphere of radius 1 and compute it’s volume, the volume will increase as you add dimensions UNTIL you get to about 6 dimensions, whereupon it reverses and heads back to zero! A sphere of radius 1 has a bigger volume in 3D than it does in 1 million-D.
What about the volume of a sphere with radius 2? Before, with R=1, the R^D portion of the volume equation did not contribute (recall that one raised to any real exponent is still one). Now, if R=2, the volume increases with increasing dimension (edit: provided the gamma in the denominator does not trounce the numerator). A sphere of radius 2 has a much much much smaller volume in 3D than it does in 20D!
This is crazy when you think about it. Adding dimensions (once you get past 5) to a sphere of radius one, or less than one, actually makes its volume smaller. Adding dimensions, up to a point, to a sphere of radius greater than one makes its volume larger. This is not intuitive to our 3D minds. We think you can fit more “stuff” into a 3D sphere than into a 2D circle, provided the radius is the same. The math says this idea does not generalize. Adding more dimensions to the unit sphere makes its volume smaller!