There is no mystery here. If you pick a random real number and find the best quality approximation of it — as calculated by log_b 1/|x-a/b| — then the median quality of the best such approximations within a tolerance of 1/1024 is around 3.19. So with respect to the measure this article is talking about, pi is completely normal and 355/113 is a completely unexceptional best approximation with a denominator of at most nine binary digits (about 3 decimal digits). 355/113 isn't even the best quality approximation of pi by the given criterion — that would be 22/7, which, as mentioned in the article, has a quality measure of 3.429288337281781.
These kinds of articles about pi, or numbers, where they try to summon up some mystery and wonder around it, come across to me as not very genuine. I get the impression it gets upvoted because people want to feel like they're smarter, or more geeky and hip for having looked at it.
One of the most "mind boggling" areas of math is number theory
Math dealing with real/complex numbers has been vastly explored, and there are very powerful tools, and computers are very good at these kind of problems (Calculus, etc)
With number theory, progress is much harder, things work in a completely different logic (think for example that 5+3 can be 0 for example)
I don't know, I've met folks for whom math is so magical they do nothing but talk about it all the time. I can recognize that passion in other pursuits as well, from knitting to physics.
I find a close correlation between people who play with numbers to people who like to solve puzzles.
Enumerate all the fractions with at most 3-digit denominator (0, 1, 1/2, 1/3, 2/3, 1/4, ..., 998/999) between 0 and 1. Now what is the probability that amongst them we find an approximation to a random real that is at least as good as 355/113 is to pi?
Seems to be about 15%, so not much mystery here. But we allow denominators as large as 999 because we use the decimal system. In hexadecimal it would be at most 2-digit denominators, up to ff = 255.
In this case, the probability is about 1%. Much more mystery. So it comes down to, how large a set of denominators we think 113 is drawn from, and whether living in a 1 in 100 universe is enough to be mysterious in this case.
So just a note, you've got a great deal of repetition in your search. 0.5, for instance, turns up once for every even denominator. That's going to skew your percentages a bit.
Yes, the constructed list contains a lot of duplicates (removing duplicates with the python set operations shortens it by 40% - would be a worthwhile optimization for the runtime, too), but my percentages are not based on that.
But: I sampled a random real, and looked at whether any fraction in the list is close enough to this real. And repeated the sampling 1000 times. This is how the percentages are calculated, and the duplicates in the list don't affect this.
Julia code to figure this out can be found here:
https://gist.github.com/3170899