Interesting article. The difference between the square root of two and pi sequences is surprising to me; is there some intuitive reason why pi looks so much more regular?
BTW, you probably never found the irrationality measure to be very useful, because it's well known that almost all real numbers have irrationality measure 2.
For a more comprehensive explanation of this behavior, look into the subject of Diophantine approximation; root 2, phi and other quadratic irrationals are badly approximable numbers (phi being the king of those), and their ability to "evade" rationals for a long time means their multiples are free to scatter around the interval more flexibly than better approximable ones. Diophantine approximation formalizes this heuristic; a good introduction is Khinchin's classic Continued Fractions.
In a similar vein there is there’s this video explaining why phi is somehow “more irrational” than other numbers. It looks like a special case of the same kind of analysis you are talking about
if there's a good rational approximation to some number x = n/m + ε where n and m are integers and ε is small, then every m multiples you get back to almost the same place; only the residual difference ε remains. so you get m clusters of points that are each spread apart by ε. after enough multiples, these clusters grow big enough to overlap, and then you start to see clusters corresponding to the next approximant
the first few approximants of pi using http://canonical.org/~kragen/sw/netbook-misc-devel/contfrac.... are 22/7, 333/106, 355/113, and 103993/33102. the large jump from 22/7 to 333/106 means that ε is tiny in π = 22/7 + ε. so you see 7 cleanly defined groups, and that's all you'll see for a long time, until you have several hundred samples, at which point i think you'll have 106 or 113 clusters, depending on how you squint, and that will remain the case for tens of thousands of samples
by contrast the first few approximants of √2 are 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, etc. you never get a large jump from one denominator to the next
btw does anyone know how to wring these out of pari/gp? vecsort(vecsort([bestappr(Pi, i) | i <- [1..10000]],,8), (a, b) -> denominator(a) - denominator(b)) gave me [3, 13/4, 16/5, 19/6, 22/7, 333/106, 355/113], and i don't really know what to make of that. also it's obviously not a good way to get approximants like π ≈ 4272943/1360120. i don't know how to use pari/gp very well
Not directly related to the phenomenon in the article, Numberphile has a video[1] that goes into why pi has a surprising amount of regularity despite being transcendental.
Writing square root of two as a continuous fraction leads to a periodic sequence of coefficients. More generally, a continuous fraction is periodic if and only if the number it represents is a quadratic number, that is, a linear combination of a rational number and the square root of a rational number [1]. It does not exactly answer your question, yet in the sense of continuous fraction at least, square root of two is more structured than pi, and less random.
Edit: the golden ratio is also a quadratic number, so this intuition is wrong in the end!
