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.
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?
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.
> "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...
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.
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.