winding graphs around circles

January 29, 2018 6:19 AM

But what is the Fourier Transform? A visual introduction. (previously)
posted by kliuless (41 comments total)

Fourier is really an underappreciated Autobot.
posted by leotrotsky at 6:56 AM on January 29, 2018 [4 favorites]

tho actually a better representation of the Fourier transform would be Voltron disassembling into its constituent robot lions.
posted by leotrotsky at 6:58 AM on January 29, 2018 [8 favorites]

I do not miss the upper maths I took for my Engineering Degree. I always felt like there was amazing stuff lurking about but I had so much other stuff on my plate, like a heavy course load in general my last few years, passing Heat Transfer and managing a Senior Design Project, not to mention that learning maths like this from folks who, quite literally, spoke very limited English (and only with another country's English accent layered on top of their native accent) that it was just more of a 'get it done for the love of God' situation rather than a deep dive that I might have enjoyed.

Props to videos like this that can explain things a bit more clearly.
posted by RolandOfEld at 7:00 AM on January 29, 2018 [7 favorites]

3 blue 1 brown is my favorite math Youtube channel, even above numberphile, because numberphile is more "Gee, that's neat" whereas 3 blue 1 brown is more, "Now you will UNDERSTAND eigenvalues."

That's pretty wicked. One of my life's biggest regrets is too little math, too late.

Having a broad understanding of higher mathematics is like the closest thing to learning magic that exists in the world. You suddenly have this skeleton key that unlocks all sorts of cool stuff across all the disciplines of human endeavor.
posted by leotrotsky at 7:05 AM on January 29, 2018 [29 favorites]

Can't wait to watch this. The Fourier transform was one of my favourite things that I learned in college.
posted by tobascodagama at 7:08 AM on January 29, 2018 [2 favorites]

Wow. I definitely remember being exposed to the calculation itself at several points in college and grad school (mostly in p. chem I think, and/or in some of the math class prerequisites to p. chem), at a time in my life when the performing of integrals of imaginary exponentials didn't feel weird or especially difficult - but never *intuitive*. Despite pretty much every math and physics professor insisting it was very much an intuitive function, never was anyone able to provide me that straightforward understanding of what the heck it was doing. This video is fantastic.
posted by solotoro at 7:18 AM on January 29, 2018 [5 favorites]

Fourier transforms made me fail 2nd year Engineering Math 3 times before I could graduate half a year late.
posted by infini at 7:34 AM on January 29, 2018 [3 favorites]

The Fourier transform exists so undergrads can still learn something if they fail to grasp lapalace transforms.
posted by GuyZero at 7:44 AM on January 29, 2018 [9 favorites]

iirc both were useful for my vibrations course
posted by indubitable at 7:50 AM on January 29, 2018 [1 favorite]

First year audiology school we had to do these by hand, build DFTs from scratch in matlab, and then finally just use the built-in matlab FFT function. It was painful at the time but useful in understanding FTs, and now the FT is the central function of my working life.

In any case, this was awesome. Thanks for posting!
posted by Lutoslawski at 7:59 AM on January 29, 2018 [1 favorite]

I've been trying to help my coworkers build their intuitions about time-frequency analysis, so this is perfect timing (but indeterminate frequency ha ha ha). Thanks for posting!
posted by biogeo at 8:22 AM on January 29, 2018 [3 favorites]

That was the clearest explanation of Fourier Transforms that I've ever seen. College professors I've had never seemed interested in explanation the physical explanations of a lot of the tools we needed, much to the detriment of understanding I think. Derivations of equations generally were presented in one of two ways:

1) Start with an expression. "Well, let's see what happens when we do this to it..." Many pages later, "Voila! We have stumbled upon Navier Stokes!"

2) Start with a question. "Well, let's assume that the answer has a solution of this form." Magic happens. "We have proven that the solution has the form that we assumed!"

I always felt like these approaches are terrible for learning math because they give the impression that all these intelligent people who have equations named after them were... guessing, basically. There's never any relating back to the physical world (which, granted, isn't always possible), and I think that makes it difficult to apply the learning to new problems.
posted by backseatpilot at 8:33 AM on January 29, 2018 [18 favorites]

This was wonderful for this maths-challenged idiot.
posted by maxwelton at 8:41 AM on January 29, 2018

I don't know if it's necessary to actually relate anything back to the physical world (especially with something like Fourier transforms), but I do think that proofs could be presented with more historical background. Like, what was Fourier trying to do that required this invention of this new method?

When you try to kludge a "physical" analogy on top of every mathematical concept, you get weird stuff like my electromagnetics professor's "Can you tell me the flux of... a grasshopper?" (I loved the guy, he was an excellent teacher, but some of his analogies were just bizarre.)
posted by tobascodagama at 8:51 AM on January 29, 2018 [3 favorites]

Yeah, that's a better way of putting my feelings - context is important! You almost need a parallel "history of math" course alongside the technical learning. Another excellent reason to encourage more liberal arts education for STEM students.
posted by backseatpilot at 9:00 AM on January 29, 2018 [8 favorites]

2) Start with a question. "Well, let's assume that the answer has a solution of this form." Magic happens. "We have proven that the solution has the form that we assumed!"

This one in particular always drove me up the wall in differential equations. Looking back it would have made sense to just shove a mechanical solution at you to memorize and move on if it were a class just for engineers, but it was a general math course that included people who supposedly study the field for deeper understanding, but none was to be found. I guess they covered that stuff in a course just for math majors, maybe one of the analysis ones?
posted by indubitable at 9:29 AM on January 29, 2018 [1 favorite]

OK so the curve wrapping thing: the y-axis of the sinusouid is reperesented by distance from the origin?
posted by thelonius at 9:40 AM on January 29, 2018

Yep. Note, though, that in their depiction the original waveform (the sinusoid, or combinations thereof) has been shifted up so that it bottoms out at 0; the y-axis values are all non-negative. So when the original curve hits the bottom of the trough, the wrapped curve hits the origin, and everywhere else the wrapped curve has a positive distance from the origin.
posted by Westringia F. at 9:56 AM on January 29, 2018 [1 favorite]

I just watched this over the weekend and was again blown away at how lucid and clear all of his videos are. I would have killed to take classes under him in school. My first year of calculus was taught by sullen math grad students for whom it was obvious they'd rather be doing any thing else. I learned way more from my calculus-based physics class that I was taking concurrently taught by a prof that seemed to actually enjoy teaching.

Bill Hammack has a neat series on Michelson's harmonic analyzer that was used to mechanically compute Fourier transforms and is neat in its own right, but also gives some interesting mechanical analogy to the computation process.
posted by Dr. Twist at 10:19 AM on January 29, 2018 [2 favorites]

So while I was watching this, I suddenly remembered the engineerguy's playlist about the mechanical fourier transformation machine, which caused a moment of realization that frickin' _hurt_ while watching 3blue1brown's video:

https://www.youtube.com/watch?v=NAsM30MAHLg

I totally agree, though, about how much context matters. All of my high school science courses were basically recapitulating the history of that science and how they learned what they learned and _why_ they learned what they learned. Math was always sort of presented more as something handed down from on high, which explains why I struggled with it.
posted by Kyol at 10:22 AM on January 29, 2018 [2 favorites]

I read something a few months back - I think it was in the introduction to a math textbook - which talked about an expectation that university math students "show more independence" in their learning than high school students. That's why there were few intuitions or examples or extended explanations; as math students matured as mathematicians, they were expected to figure more and more of that out for themselves. The bare equations, my friends, should be all you need.

From what I've read, the Bourbaki group took that attitude to the extreme. I suspect that I'm too lazy for that ascetic approach to work for me.
posted by clawsoon at 10:47 AM on January 29, 2018

FFTs helped grease my exit from CWRU in 1982. Although the prof who taught the class was nice.
posted by lagomorphius at 12:45 PM on January 29, 2018

I think I understood that. Thanks Kliules!
posted by stanf at 1:41 PM on January 29, 2018

OK, next do Laplace!
posted by notsnot at 1:55 PM on January 29, 2018 [2 favorites]

My first job after uni, I worked on a non-destructive test rig which used ultrasonic sound waves to probe metal for defects. If the material was consistent, the sound waves would reflect off the back wall "cleanly" but if there were holes or different density, this would cause a ringing at a specific frequency, so we used an FFT which I ended up coding in 80387 assembly - that wasn't fun.
posted by parki at 2:13 PM on January 29, 2018 [5 favorites]

My first year of calculus was taught by sullen math grad students for whom it was obvious they'd rather be doing any thing else.

Oh God fuck those world-weary, self-important 21 year olds. Some of those lab TAs were among the few people I have actually wanted to hurt physically.
posted by thelonius at 2:18 PM on January 29, 2018

Good video but I'm confused about how we go from the g hat function definition (with the integral) to work out the large spikes?
Does that require differentiation wrt frequency?
posted by 92_elements at 2:23 PM on January 29, 2018 [1 favorite]

A while ago I translated part of an engineering thesis which talked a lot about Fourier transforms, so maybe I'll finally get to understand what the hell it was about!
posted by lollymccatburglar at 2:45 PM on January 29, 2018 [1 favorite]

I took several years of calc but no further and just knew the name Fourier transforms and that is all. I’m 48 and had a tiring day and ended up watching all 20 minutes. I did get a bit lost 14 minutes in since my last encounter with e was like 30 years ago, but this was great.

I had no fucking idea about the puzzler at the end though. Math word salad, help!
posted by freecellwizard at 6:51 PM on January 29, 2018 [1 favorite]

my last encounter with e was like 30 years ago

e ain't nothing but a number
posted by thelonius at 6:57 PM on January 29, 2018 [4 favorites]

The bare equations, my friends, should be all you need.

I like to use poetry as an analogy. Think of all that math stuff as poetry written in some language. Do you sit down with a dictionary and grammar and slowly work out what a poem means? Of course not -- you know the language and you can appreciate the meaning (and beauty) of the poem as you read it. And you may have to read it a few times to fully appreciate its meaning.

In other words, while you're still learning this "math" language, you work with examples and try to develop the right intuition. (How do you learn a language? Practice, practice, practice. And watch all the other videos by 3B1B, as well as those by Welch Labs.) By the time you get to university you are expected to know the language.

One of my favourite inspirational quotes is in one of 3B1B's videos, "Mathematics is about proving theorems like poetry is about writing sentences." (paraphrased, since right now I can find neither the quote nor the video.)
posted by phliar at 7:23 PM on January 29, 2018

> I had no fucking idea about the puzzler at the end though. Math word salad, help!

Oh, the puzzler at the end has nothing at all to do with the rest of the video! (Which is a pity, because it would have been fun for him to cap off this very cool video with a question that would have kept the viewer thinking about it...)

Anyway, what the puzzler is asking is this: suppose you have a 3-d blob of points that's convex, ie, it doesn't have any dimples or divots. (Or, to put it a bit more rigorously, if you picked any two points on the surface and connected them by a straight line, that line would always be on the inside of the blob.) You can describe any point on the surface of said blob as a vector from the origin to that spot. Now, suppose you took two spots, and added their vectors together; what you'd get is a vector pointing to a new spot in space. You could do this for all possible pairs of points on the original surface, and the points you'd get would form a whole new blob. He's asking you to prove that the new blob is also convex, ie, dimple-free.

[More formally, he's asking about the Minkowski sum of the original set and itself.]
posted by Westringia F. at 8:23 PM on January 29, 2018 [3 favorites]

I like to use poetry as an analogy. Think of all that math stuff as poetry written in some language.

This makes a lot of sense. I can't read poetry, either, and for much the same reason, but that similarity had never occurred to me before. I can read through a piece readily enough, recognizing all the pieces and grasping their meaning on a fine-grained level, but when I reach the end, I generally have little or no idea what the piece as a whole was trying to say.
posted by crotchety old git at 7:36 PM on January 30, 2018 [1 favorite]

My biggest shock in math was moving from an American high school wiht AP Calc to a Bangalore engineering college with Russian textbooks for Calc. Teh shock never wore off but I scraped myself into a First Class degree by acing the optimisation and the ops research and the Fortran.
posted by infini at 3:11 AM on January 31, 2018

No wait, the year before (1982) the shock was O Level Math to American disaggregated math. I'm just a puddle of mathshock even now.
posted by infini at 3:12 AM on January 31, 2018 [1 favorite]

I want to spend $5 on Puddle of MathShock Even Now
posted by infini at 3:13 AM on January 31, 2018 [2 favorites]

my last encounter with e was like 30 years ago

I tried it couple years ago for the first time in a decade but I had a really bad experience coming down.
posted by atoxyl at 11:00 AM on January 31, 2018 [2 favorites]

Street e now is all cooked up in backyard math labs
posted by benzenedream at 10:39 PM on February 1, 2018 [3 favorites]

Yeah plus it might not have been real e at all -- it has so many derivatives.

Everybody says they're exactly the same as the original, but I have my doubts.
posted by jamjam at 12:40 AM on February 2, 2018 [6 favorites]

Street math is why I have to show my id everytime I buy sinusoid
posted by condour75 at 7:35 AM on February 4, 2018 [1 favorite]

Having a broad understanding of higher mathematics is like the closest thing to learning magic that exists in the world.

my working assumption is that math is the only true theology.

I like to use poetry as an analogy. Think of all that math stuff as poetry written in some language.

-binding the andat!
-Vladimir Voevodsky, 1966 — 2017
-A Categorical Semantics for Causal Structure
-Split Octonions and the Rolling Ball, Dr. John Baez :P
posted by kliuless at 9:19 PM on February 8, 2018