3 ms·

Squinting at this, I wonder if it's at all valid to say that the existence of a quadratic time algorithm to calculate pi has anything to do with the fact that t

Squinting at this, I wonder if it's at all valid to say that the existence of a quadratic time algorithm to calculate pi has anything to do with the fact that the implicit formula of a circle is made up of quadratic terms. In other words, if pi basically sums up the most important fact about a circle's geometry, then it's reasonable to expect that geometry to be represented somehow in the important facts about algorithms that calculate pi.

alanbernstein 3y ago
That's an interesting concept. I think similar spigot algorithms are known for other transcendentals, and I suspect if you compared them you would not find a general trend of deep connections between algorithmic complexity and the geometric features of the corresponding value. What would you look for in the spigot algorithm for e, or log 2?
- davesque 3y ago
  I suppose e's connection to hyperbolic geometry might suggest a relationship with the implicit formula x^2 - y^2 = 1. And I guess log2's behavior would be very much connected to that since it only differs from the natural log by a constant factor.
  - davesque 3y ago
    But I know I'm reaching here. I just like fantasizing about math :).
    - westurner 3y ago
      Is there unitarity, symmetry, or conservation when x^2 ± y^2 = 1? We square complex amplitudes to make them real. https://twitter.com/westurner/status/967970148509503488 https://twitter.com/westurner/status/967970148509503488 : > "Partly because, mathematically, wavefunctions are vectors in a L^2 Hilbert space, which is complex-valued. Squaring the amplitude, rather Ψ∗Ψ=|Ψ|^2 is one way to ensure that you get real-valued probabilities, which is also related to the fact that […]" https://physics.stackexchange.com/questions/280748/why-do-we-square-probability-amplitude-why-not-direct-values#comment630152_280748 https://physics.stackexchange.com/questions/280748/why-do-we...
  - eru 3y ago
    Well, that constant factor is irrational..
aquafox 3y ago
There are different ways to define pi. In real analysis, you first define the exponential function: exp(x) = sum (x^k)/(k!), then cosine: cos(x) = Re(exp(i*x)), where i is the imaginary unit, and then show that cos(x) has exactly one root in [0,2], which you call pi/2. Your statement suggests that the definition via the circle is more fundamental that other definitions, which it isn't, e.g. because it requires a very special metric in Euclidean space (of which there are infinitely many), while real analysis only requires a metric on the real numbers.
- deleted 3y ago
  [deleted]

alanbernstein 3y ago

That's an interesting concept. I think similar spigot algorithms are known for other transcendentals, and I suspect if you compared them you would not find a general trend of deep connections between algorithmic complexity and the geometric features of the corresponding value. What would you look for in the spigot algorithm for e, or log 2?

davesque 3y ago
I suppose e's connection to hyperbolic geometry might suggest a relationship with the implicit formula x^2 - y^2 = 1. And I guess log2's behavior would be very much connected to that since it only differs from the natural log by a constant factor.
- davesque 3y ago
  But I know I'm reaching here. I just like fantasizing about math :).
  - westurner 3y ago
    Is there unitarity, symmetry, or conservation when x^2 ± y^2 = 1? We square complex amplitudes to make them real. https://twitter.com/westurner/status/967970148509503488 https://twitter.com/westurner/status/967970148509503488 : > "Partly because, mathematically, wavefunctions are vectors in a L^2 Hilbert space, which is complex-valued. Squaring the amplitude, rather Ψ∗Ψ=|Ψ|^2 is one way to ensure that you get real-valued probabilities, which is also related to the fact that […]" https://physics.stackexchange.com/questions/280748/why-do-we-square-probability-amplitude-why-not-direct-values#comment630152_280748 https://physics.stackexchange.com/questions/280748/why-do-we...
- eru 3y ago
  Well, that constant factor is irrational..

aquafox 3y ago

There are different ways to define pi. In real analysis, you first define the exponential function: exp(x) = sum (x^k)/(k!), then cosine: cos(x) = Re(exp(i*x)), where i is the imaginary unit, and then show that cos(x) has exactly one root in [0,2], which you call pi/2. Your statement suggests that the definition via the circle is more fundamental that other definitions, which it isn't, e.g. because it requires a very special metric in Euclidean space (of which there are infinitely many), while real analysis only requires a metric on the real numbers.

deleted 3y ago
[deleted]