I can only infer from the picture of a person on the couch two-thirds of the way through the article that this is about "Prof. Manjul Bhargava": https://en.wikipedia.org/wiki/Manjul_Bhargava
> After the talk about “Poetry, Drumming and Mathematics”, we arrived at the appointed time at the Bose–Einstein Guest House to find our man, with his mother, standing transfixed by the deer grazing on the lawns. We managed to prise him away for a while — here’s what followed.
is meant to provide that intro/context; but, like you and the grandparent, I was puzzled about who 'our man' was.
Manjul makes it so simple. The odds of two numbers being relatively prime is 6/pi^2 so where's the circle? You can kind of see rotation symmetry if you draw the relatively prime pairs of numbers in the coordinate plane. However that symmetry is far from perfect.
Have a look at https://en.wikipedia.org/wiki/Basel_problem . I'm not a mathematician, and I'm probably wrong, but I think it has to do with infinite series that "circle" around a value, and asymptotically approach a value, only accurately expressed in terms of pi.
"Euler's original derivation of the value π2/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series."
Sorry, not very convincing or geometric, and I'm sure someone else can provide a better answer, but that's how I visualize it.
A simple explanation I enjoy (not quite visual, but geometric in some sense) is the on the wiki using Fourier series, understanding it requires just a few steps:
1) You accept Parseval's identity: by changing between orthogonal basis, the energy (sum of absolute squared values) remains the same.
2) You find that the energy of the periodic function x mod pi is sum(1/n^2).
3) Evaluating the integral gives pi^2/6!
The square is neatly explained because it involves energies. I guess this is why the pattern works only for even powers.
