Comment by cubefox
Also interesting:
The area of the Euclidean unit disk (area of a circle with radius 1) is equal to π. However, the volume of the unit ball (ball with radius 1, one dimension more) is larger, namely 4/3π ≈ 4.18879. So how does the hypervolume for unit n-balls change in higher dimensions, where the 1-ball is a circular disk and the 2-ball an ordinary ball?
Surprisingly, it first increases but then converges to 0. The maximum is achieved for a unit 5-ball with a hypervolume of about 5.2638. For higher dimensions the value decreases again.
However: If we allow fractional dimensions, the 5-ball isn't at the peak volume. The n-ball with the largest volume is achieved for n≈5.256946404860577, with a volume of approximately 5.277768021113401, which are slightly larger numbers.
These were computed by GPT-5-thinking, so take it with a grain of salt. But the fractional dimension for peak volume is also reported here on page 34: http://lib.ysu.am/disciplines_bk/8d6a1692e567ede24330d574ac3...
Curiously, the paper above says that the area of the hyper surface of the n-ball (rather than its volume) peaks at n≈7.2569464048, while ChatGPT calculated it as n≈6.256946404860577, so exactly one dimension less than the paper. I assume the paper is right?
Also curiously, as you can see from these numbers, that fractional dimension with the peak hyper surface area is exactly two (according to the paper) or one (according to ChatGPT) dimension larger than the fractional dimension of the peak volume.
This is a nice analog but unfortunately I think it breaks down in a way the "π" calculation does not.
In the article 2π(d) = the ratio of the circumference to the radius. This is dimensionless, in the sense that the circumference and the radius are both lengths (measured in meters, or whatever), so 2π(d) is really just a number.
But the (hyper)volumes you're talking about depend on dimension, which is exactly why you say "hyper". In 2 dimensions the volume is the area, πr^2, which has dimensions L^2 [measured in m^2 or whatever]. But in 3 dimensions the volume is 4/3 πr^3, which has dimensions L^3. The 5 dimensional (hyper)volume has dimensions L^5, and so on.
So, "comparing" these to find out which is bigger and which smaller is not really meaningful---just like you shouldn't ask which is the bigger mass: a meter or a second? Neither is, they aren't masses.