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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 t

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. http://ocw.mit.edu/courses/mathematics/18-786-topics-in-algebraic-number-theory-spring-2010/18-786s10.jpg http://ocw.mit.edu/courses/mathematics/18-786-topics-in-alge...

impendia 11y ago
Yeah, but the rotational symmetry doesn't explain why the pi is there. (e.g. why is it squared?) If you can explain, convincingly and geometrically, why pi is present in this formula then I would be VERY VERY interested to read.
- Intermernet 11y ago
  Have a look at https://en.wikipedia.org/wiki/Basel_problem 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.
  - impendia 11y ago
    No apologies necessary, I think that's just the state of mathematics. The sum of the inverse fourth powers is pi^4/90. Sixth powers, pi^6/915. The pattern keeps going for even powers and not for odd. I don't think there is a geometric proof. But if there were, it would be incredible.
    - darkmighty 11y ago
      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.

impendia 11y ago

Yeah, but the rotational symmetry doesn't explain why the pi is there. (e.g. why is it squared?) If you can explain, convincingly and geometrically, why pi is present in this formula then I would be VERY VERY interested to read.

Intermernet 11y ago
Have a look at https://en.wikipedia.org/wiki/Basel_problem 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.
- impendia 11y ago
  No apologies necessary, I think that's just the state of mathematics. The sum of the inverse fourth powers is pi^4/90. Sixth powers, pi^6/915. The pattern keeps going for even powers and not for odd. I don't think there is a geometric proof. But if there were, it would be incredible.
  - darkmighty 11y ago
    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.