Complex Math Made Simple With Engaging Animations: Fourier Transform, Calculus, Linear Algebra, Neural Networks & More

In many an audio engi­neer­ing course, I’ve come across the Fouri­er Trans­form, an idea so fun­da­men­tal in sound pro­duc­tion that it seems essen­tial for every­one to know it. My lim­it­ed under­stand­ing was, you might say, func­tion­al. It’s some kind of math­e­mat­i­cal reverse engi­neer­ing machine that turns wave­forms into fre­quen­cies, right? Yes, but it’s much more than that. The idea can seem over­whelm­ing to the non-math­e­mat­i­cal­ly-inclined among us.

The Fouri­er Trans­form, named for French math­e­mati­cian and physi­cist Jean-Bap­tiste Joseph Fouri­er, “decom­pos­es” any wave form into fre­quen­cies, and “vir­tu­al­ly every­thing in the world can be described via a wave­form,” writes one intro­duc­tion to the the­o­ry. That includes not only sounds but “elec­tro­mag­net­ic fields, the ele­va­tion of a hill ver­sus loca­tion… the price of  your favorite stock ver­sus time,” the sig­nals of an MRI scan­ner.

The con­cept “extends well beyond sound and fre­quen­cy into many dis­parate areas of math and even physics. It is crazy just how ubiq­ui­tous this idea is,” notes the 3Blue1Brown video above, one of dozens of ani­mat­ed explo­rations of math­e­mat­i­cal con­cepts. I know far more than I did yes­ter­day thanks to this com­pre­hen­sive ani­mat­ed lec­ture. Even if it all seems old hat to you, “there is some­thing fun and enrich­ing,” the video assures us, “about see­ing what all of its com­po­nents look like.”

Things get com­pli­cat­ed rather quick­ly when we get into the dense equa­tions, but the video illus­trates every for­mu­la with graphs that trans­form the num­bers into mean­ing­ful mov­ing images.

3Blue1Brown, a project of for­mer Khan Acad­e­my fel­low Grant Sander­son, has done the same for dozens of STEM con­cepts, includ­ing such sub­jects as high­er dimen­sions, cryp­tocur­ren­cies, machine learn­ing, and neur­al net­works and essen­tials of cal­cu­lus and lin­ear alge­bra like the deriv­a­tive para­dox and “Vec­tors, what even are they?”

In short­er lessons, you can learn to count to 1000 on two hands, or, just below, learn what it feels like to invent math. (It feels weird at first.)

Sander­son­’s short cours­es “tend to fall into one of two cat­e­gories,” he writes: top­ics “peo­ple might be seek­ing out,” like many of those men­tioned above, and “prob­lems in math which many peo­ple may not have heard of, and which seem real­ly hard at first, but where some shift in per­spec­tive makes it both doable and beau­ti­ful.” These puz­zles with ele­gant­ly clever solu­tions can be found here. Whether you’re a hard­core math-head or not, you’ll find Sanderson’s series of 3Blue1Brown ani­ma­tions illu­mi­nat­ing. Find them all here.

Relat­ed Con­tent:

Free Online Math Cours­es

The Map of Math­e­mat­ics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

Cit­i­zen Maths: A Free Online Course That Teach­es Adults the Math They Missed in High School

Free Math Text­books 

Math Mag­ic

Josh Jones is a writer and musi­cian based in Durham, NC. Fol­low him at @jdmagness

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  • Dave Dietrich says:

    I enjoyed the arti­cle and video on Fouri­er Trans­forms. I applied FFTs in my new the­o­ry on mechan­i­cal clock repair diag­nos­tics, and believe you can apply to any­thing mechan­i­cal with spin­ning parts. e.g., you can hold a micro­phone near a car’s engine and decom­pose each part to see how ‘loud’ it is vs. spec. My clock video:

    Thanks again,
    Dave Diet­rich

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