What particularly struck me about the article is that I want to apply Benford's law to pi, see if it works, and if it does not, determine if there is variance at different lengths. Then try it on pi*r^2 ...
Try it out! It's probably a five line script. (I'd like to, but I can't justify doing a fun programming problem that's not work right now.)
Benford's Law only applies to leading digits, and only for numbers following a certain class of distributions.
It's important to note that the "Law" is not an innate mathematical property of anything, but an observed phenomenon with mathematical underpinnings. It doesn't apply exactly to any well-defined, real-world distribution ("physical constants" is not a well-defined distribution) but it applies approximately to a good number of them.
Benford's Law is exactly true on 10^X, where X is a random variable, chosen with uniform probability, from [0, 1). It's also true with X chosen from [0, n) for any integer n, because the leading digit of 10^X relies only on the non-integral part of X. When X is chosen from some other distribution that spans many orders of magnitude (say, N(4, 1)) the distribution of the fractional part (X - floor(X)) is approximately a uniform random variable from [0, 1), so Benford's Law is approximately true.
This is why I find the concept interesting. I want to see the conjectures and theories /in person./ I love mathematical theory, in general, but my knowledge is very shallow. I more-or-less understand the properties of pi that you described, but I want to test them. I also want to test the properties of Benford's law to see how they work and how they can or can't be applied. This looks like the perfect opportunity to do exactly that.