Why Pi Matters (2015)
newyorker.com
newyorker.com
There are also good reasons to switch to Tau (τ=2π):
| https://tauday.com/tau-manifesto
But then, there is also Radical Tauism, proposing to switch to √(2π):
| https://cp4space.wordpress.com/2013/02/28/radical-tauism/
And of course there is the famous XKCD compromise on that matter:
When people describe sines and cosines by breaking a full revolution down into 2π radians, they don't do it for no reason. They do it because this manifests itself loudly in the differential properties of sines and cosines.
Fourier transform works perfectly OK if you redefine sine and cosine functions to have different periods. That's basically equivalent to a variable substitution in the integral. As far as I can see, the transform itself doesn't favor any specific definition and pi has no special connection to periodic functions like the article implies.
It's also true that sine and cosine functions are most conveniently defined in terms of radians, which is what I think Chinjut is trying to say. But that is more or less unrelated to the Fourier transform.
Fourier series of functions from R/Z to C (i.e., functions of period 1; i.e., functions f from R to C such that f(x + 1) = f(x) for all x) uses specifically cos(2πNx) and sin(2πNx) for integer N; you can't replace 2π here by any other value (since then you don't get the set of cosines and sines of period 1).
That's the special connection between 2π and periodic functions: every function of period 1 decomposes into the sum of a series of functions whose 2nd derivatives are square numbers * -(2π)^2 * themselves. [Or, more cleanly, into the sum of a series of functions whose 1st derivatives are integers * 2πi * themselves]. Again, you cannot replace 2π with any other value in the statements of this paragraph (well, except its negation...).
Define cos'(x) = cos(2πx/A) and sin'(x) = sin(2πx/A) (such functions could be defined without referring to π in their definition)
Now f can be expressed as a weighted sum of cos'(ANx) and sin'(ANx) for integer N.
> every function of period 1 decomposes into the sum of a series of functions whose 2nd derivatives are square numbers * -(2π)^2 * themselves
You are correct that the scaled sin and cos functions I mention above still have the 2nd derivative equal to -(2π)^2 * themselves.
I found that a weak argument that pi is somehow inherent in periodic functions though.
First, this property of sin and cos is not necessary for or directly connected to the Fourier transform as far as I can see.
Second, there are infinitely many families of orthogonal functions that can be used to decompose periodic functions in the same way sin and cos can be used and do not fit this rule of the 2nd derivative. Consider a family of square wave functions for instance.
π (2π, etc.) truly is special for the Fourier transform: it is inherent in decomposing a periodic function into a sum of exponentials (and cosine and sine are just even and odd components of exponential functions). An exponential function from R to C is of period 1 if and only if the natural logarithm of the base of the exponential is an integer multiple of 2πi (i.e., just in case the function's derivative is an integer multiple of 2πi times itself).
It's true that there are other (non-exponential) families of orthogonal functions that can be used for decompositions, but the fact that the Fourier transform is specifically in terms of a family of exponential functions is of some significance (for example, it means multiplication on one side of the transform corresponds to convolution on the other side; it's also what makes the Fourier transform and inverse Fourier transform essentially the same operation).
Tl;dr: Periodic functions can be represented in various ways, but the Fourier series decomposition is a particularly useful representation because it "diagonalizes" differentiation (which is to say, it "diagonalizes" translation by tiny (and consequently also by arbitrarily sized) amounts). And what we discover in diagonalizing the differentiation operator on functions of a fixed period is that its eigenvalues are precisely the integer multiples of 2πi divided by the period. This is a fundamental connection between π and the differential(/integral/etc.) structure of periodic functions.
But I was actually quite serious. Prime chaos is the result of taking additive modularity and relaying it to multiplicative modularity. Form creating function. Function creating form. Two things hardly related but at a junction through numbers, through counting.
If you think my mention of mirror and partitioning has pangs of insanity, you should study how the gaps of primes evolve...its just a waveform defined by previous primes. And yes, it does involve mirroring and partitioning. Primes are just an easy target for bullshit if you dont even understand what belies each prime generating a waveform that then causes interference with the existing sieve waveform (that belies all previous primes.)
If this interests you or you have any questions, I'd be happy to do a skype or hangouts call to show you how primes are actually very analagous to 'an endless convoluting ticker.'