
Why Pi Matters (2015) - Osiris30
http://www.newyorker.com/tech/elements/pi-day-why-pi-matters?currentPage=all
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meta_AU
Pi is too small. We should just move to Tau and make things easier.

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krylon
Tau is too large. We should settle on Pi * 1.5 (or Tau / 1.5), thus making
life ... interesting for both Pi lovers and Tau lovers.

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FreeFull
I object to your use of the arithmetic mean. We should actually settle on Pi *
sqrt(2)

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krylon
Yes, that idea is even better! =D

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goldenkey
Its a bit coy to call on Pi's significance in a Fourier transform where the Pi
is simply the units inside of sin and cos calls. I kind of lost a bit of joy
for the article after that gulp.

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Chinjut
What do you mean? In terms of Fourier series, the significance is that a
function of period 1 breaks down into a sum of components whose derivatives
are integer multiples of 2πi times themselves (or, if you wish to avoid
descriptions which make use of complex numbers, whose second derivatives are
square multiples of -(2π)^2 times themselves).

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.

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goldenkey
I mean that sin and cos are tied to Pi in terms of their period. The antipode
and catipode. So one might as well group all 3 quantities as Pi related. And
if you merely take sin or cos into your formula because they are continuously
differentiable periodic functions, then well, couldnt you have just taken
something else and the Pi really loses its 'sanctity'? I'm on my phone but I'm
starkly betting that there are Fourier transforms that use diff periodic
functions instead of the trig ones. Maybe computable ones instead of symbolic,
although I'd have to concede profundity if that is the case

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avian
You are correct and that point in the article caught my attention as well.

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.

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Chinjut
Fourier transform of arbitrary functions from R to C works fine with whatever
units.

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...).

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avian
> 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

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.

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Chinjut
Sure, let's say cos'(x) is the cosine of the angle corresponding to x complete
revolutions (thus, cos' is of period 1), and similarly for sin'. Then every
function of period 1 is a unique linear combination of cos'(Nx) and sin'(Nx)
for integer N. We can say this. This is essentially the same as what we were
saying already. My point isn't that π is used in the words we say when
defining cos' and sin', but that it is part of their properties, making its
presence known once we consider derivatives: the derivative of cos' is -2π *
sin', and the derivative of sin' is 2π * cos'. [Er, the primes here don't
denote derivatives, but just our newly scaled trig functions, of course]

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.

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avian
I see your point. Thanks for taking the time to discuss this.

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goldbrick
My favorite pi formula: [https://xkcd.com/179/](https://xkcd.com/179/)

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nickm12
...and the formula that is given as an example of π's importance has a tau in
it.

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plank
Would have loved a discussion on the digits of Pi and whether they have
similar properties as those discovered of prime numbers, but this story seems
a year old ...

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goldenkey
The gaps between primes are an endless ticker defined by previous primes and
convoluted by mirroring and partitioning by future primes. Of course chaos has
a pattern. It's as simple as using data as function, and functions as data.
What makes sense in one representation is consequently nonsense in a different
Head.

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adrianN
That sounds like you read too much Time Cube.

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goldenkey
I guess studying prime gaps can do that...

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.'

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mdergosits
this should have a (2015) on the title

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extramille
Einstein was born on 14 March.

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chocolateboy
s/Storgatz/Strogatz/

