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.
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.
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
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.
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...).
> 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.
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.
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 ...
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.
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.'
I should also mention I had been programming in mathematica. So when I said different Head, I just meant different type. Guess it could have sounded kooky without that context
I will respect the rules here, I'd just really like to know what they are, specifically:
Does it mean that discussing the influence of the religious views (even, or especially? of very large groups or even most popular of them) on the topics discussed is not allowed?
Does it mean that it is not allowed writing personal view on such topics?
Is the goal to have only the comments that are never downvoted?
Is the goal to have only the comments that nobody finds "offensive"?
Is it allowed for somebody who is not religious to discuss the topics from his own point of view (the point of view of a person who certainly doesn't believe in any of the religion which can feel "offended")? To compare, would you allow a homosexual to discuss any topic from his homosexual point of view, even if some religions are "offended" by mere existence of homosexuals? If you give homosexual these rights, why not giving the same to atheists? Only because even more people can get offended?
I understand that the "rules" are something that grows with the use, so even if they weren't openly written, it would be nice if they were openly accessible and readable.
(I honestly don't see where I actually offended somebody (except for expressing a different view, I actually try to discuss to the facts)).
We're not bureaucrats or legalists, and I can't imagine anything more soul-destroying than trying to make HN's rules that specific. It would drive me mad.
Instead, try to internalize the spirit of this place. Religious flamewars are the last thing we want. Arguments about the falsehood of entire religions are sure flamewars. Please just don't go there.
It wouldn't really be a surprise if someone denied your eisegetical interpretation of scripture ("in the Bible Pi [is] exactly 3"). Those creating the "sea" in the passage usually referenced to this end clearly were able to do it, there's no reason for us to consider they didn't have as accurate a definition of the ratio as anyone else.
Of note is the tradition of numerology in Hebrew texts and the assertion that the passage on the sea includes an adjustment that renders a better approximation with a fractional percentage of error.
>"Still no cigar, 3.139 is still not more a "final truth" than 3."
Were you expecting a casual observer to give a fully accurate (!) rendering of the exact measurements of an artefact; moreover were you expecting those measurements to reveal a more accurate rendering of pi when the units are "about as long as a forearm" and "about as wide as a hand"?
> Were you expecting a casual observer to give a fully accurate (!) rendering of the exact measurements of an artefact;
No, I surely wouldn't, but I would expect from all-knowing God to pass any fact that wasn't commonly known at that time as a proof of his all-knowledge (like, "there are atoms" or "the speed of light is..."). Irrational numbers or some really exceptional (for these times) approximation of some constant would also be a nice "touch". And not to put in one of 10 commandments slaves, women and donkeys together. But He surely knew his priorities.
And note that there are enough of people that really consider and study the Bible as the source of such "all-time truths."
Your tone seems quite petty. The Bible's purpose is not to be a treatise on science; it is seeking to communicate deeper truths, not the exact nature or value of Pi. Knowing that Pi is an irrational number has not removed war, greed, nor poverty.
For the Christian bible? Well, ignoring any theological points for their own sake, I suppose it's generally accepted as attempting to imply that seeking power, riches, and glory for their own sake is futile at best if not evil and wrong, and that instead the noblest path consists of approaching the world using humility, self-sacrifice, and respect for the intrinsic dignity of one's fellow man (even strangers, foreigners, and one's traditional enemies) and one ought to pursue what is right regardless of the persecution and oppression one faces for doing so.
It supports these ideas most obviously through the headline New Testament events of the Incarnation, which puts a supreme omnipotent omniscient Divine Godhead who is purportedly worthy of all glory and honor into a dirty barn with the animals as a helpless infant, and the Passion, which flogs him brutally before crucifying him as a sacrifice for what the theology represents as the crimes of others.
However, there are also attempts towards these deeper truths in books like Wisdom and Ecclesiastes:
"I undertook great projects: I built houses for myself and planted vineyards. I made gardens and parks and planted all kinds of fruit trees in them. I made reservoirs to water groves of flourishing trees. I bought male and female slaves and had other slaves who were born in my house. I also owned more herds and flocks than anyone in Jerusalem before me. I amassed silver and gold for myself, and the treasure of kings and provinces. I acquired male and female singers, and a harem as well - the delights of a man’s heart. I became greater by far than anyone in Jerusalem before me. In all this my wisdom stayed with me.
I denied myself nothing my eyes desired; I refused my heart no pleasure. My heart took delight in all my labor, and this was the reward for all my toil. Yet when I surveyed all that my hands had done and what I had toiled to achieve, everything was meaningless, a chasing after the wind; nothing was gained under the sun."
>I would expect from all-knowing God to pass any fact that wasn't commonly known at that time as a proof of his all-knowledge //
Not wanting to extend this beyond the bounds of what is appropriate on HN but a simple (and simplistic) counter-argument is that this is exactly what God does - he reveals through inspiration of various people ideas that lead to revelation. So for example giving Newton the idea that white light was composite. We by our limitations seem necessarily to be unable to cope with a conception of the entire truth of the nature of the universe and so all our understanding is always going to be shadows on the cave wall. Revelation of the full truth of any one facet seems impossible? If the Bible said "there are atoms" then you might well argue, but it doesn't mention electroweak and strong forces, etc., etc. ad infinitum.
Revelations out of context, like telling a Phoenician what the permittivity of free-space is, would almost certainly be dismissed as nonsensical wittering. Indeed that's what happened to the Ancient Greek conception (wrong as it was) of atoms. Perhaps in a millennium or so man will look down on the atomic model as limited in the same way we look on the classical elements.
Re the 10th commandment, things you shouldn't wrongly desire (ie covet); given it's an short list of examples of things not to covet I can't see that there's much significance in those things being together other than that they're examples of things that people look at (even today) and think "I want that". Donkey (chamawr, https://www.blueletterbible.org/niv/exo/20/1/t_conc_70017) is as likely chosen for the alliteration with (chamad). If I say to you "don't look at your neighbours beautiful wife, his truck [ox] or flash car [donkey], his servants or friends [the word used in Exodus can mean either] and lust after them for yourself instead" will you accuse me of considering the things desired for to be equal somehow.
Yes, yes, "god works in mysterious ways," "people aren't ready" "they can't never know the truth anyway" etc. Doesn't hold water: religious people were quite good in preserving through the centuries a lot of material that they didn't understand.
The "ten commandments" were given from the God "in stone" and it is claimed that the texts in the Bible are the preserved words received directly from the God (not to mention that Muslims claim that the whole Quran are all the direct words of Allah).
Only whatever people preserved turned out to be trivial, never "deep" for our current knowledge. And often not only trivial, but flat-out wrong (like in Islam, that semen is produced "between the backbone and the ribs," really) so much that now the religious must claim that most of that "knowledge" was not meant to be verbatim anyway.
>so much that now the religious must claim that most of that "knowledge" was not meant to be verbatim anyway //
You know that's a very recent turn to read scripture literally? The idea that figurative interpretation is somehow a reaction to scientific challenge would be easy for you to dispel with a modicum of research.
>it is claimed that the texts in the Bible are the preserved words received directly from the God //
Not by mainstream Christianity or the Bible itself (which claims scripture is inspired by God).
> You know that's a very recent turn to read scripture literally?
Yes, and if you read my first now unfortunately gray comment, I addressed exactly the literal readings as those that limit what's easy to pass in the US: I wrote that Pi "doesn't question the literal interpretation of the Bible, 6000 years and all that. So let's celebrate that innocent number." It's right there.
> >it is claimed that the texts in the Bible are the preserved words received directly from the God //
> Not by mainstream Christianity or the Bible itself
I've meant the texts of the "ten commandments" in the Bible (you've cut the first part of my sentence, it's obvious it's about the ten commandments and I've mentioned the whole Quran as the exact words as believed by Muslims(!)) not the whole Bible, of course! If you're a believer, do you consider the text of the "ten commandments" (e.g. in Deuteronomy 5:4–21) as the actual words of God or not? Were these words exactly preserved or are they already "corrupted" in the Bible?
The US has a 'Pi day' in the same way it has a Jedi holiday on May 4th, a bunch of geeks said it for a while then more people just started saying it.
As in, the US doesn't have a 'Pi day', making the rest of your post a pointless rant against something you don't like that wasn't related until you tried to relate them.
"On March 12, 2009, the U.S. House of Representatives passed a non-binding resolution (HRES 224),[4] recognizing March 14, 2009 as National Pi Day.[10]"
The easy common acceptance versus some other possibilities is still telling. I actually don't consider it "more US" than the "Black Friday" for which I've received an equivalent number of the sales/promotion mails.
Please excuse me this time for trying to support my viewpoint, even if you seem not to have expected it, I usually do try exactly that, to present the arguments in support of my viewpoint. I apologize.
> The US House also recognized Ramadan so your politically safe assertion continues to fall apart.
How? Isn't that "recognition" (inviting for a dinner there) even weaker than the Pi Day one?
I still claim (as in my grayed post above) that the "Pi Day" is one of convenient days to which it's easy to have an agreement in the US, even among the religious, and that sadly it would be harder for a lot of newer science. And that that is the main reason why on the Pi Day we have articles like the one we comment, "Why Pi matters."
