Epicycles, complex Fourier series and Homer Simpson's orbit

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Today’s video was motivated by an amazing animation of a picture of Homer Simpson being drawn using epicycles. This video is about making sense of the mathematics epicycles. Highlights include the surprising shape of the Moon’s orbit around the Sun, instructions on how you can make your own epicycle drawings, and a crash course of complex Fourier series to make sense of it all.

In the video I attribute the animation to Santiago which is a mistake. Also Ramiro told me that unlike what it said in "Ptolemy and Homer" their animation actually involved 10000 and not just 1000 epicycles.

Anderstood’s discussion of how to create epicycle drawings in Mathematica lives here:

As usual thank you very much to Danil and Marty for their help with this video.

Enjoy!

[bluebrown]

Truly fantastic, and I love that you shared the code. Time to start pointing all those requests for Fourier series to this video (and to GoldPlatedGoof's)

[apteropith]

That's probably the clearest explanation of the discrete Fourier transform I've ever seen.

[Mathologer]

Today’s video was motivated by an amazing animation by Santiago Ginnobili from pretty much exactly 10 year ago of a picture of Homer Simpson being drawn using epicycles. Please also consider checking out Santiago’s original video and leave a ‘like’; he REALLY deserves it. Same with some of the other other resources that went into this video and that I linked to in the description.
As usual, if you’d like to support me in making these videos please consider contributing subtitles in your native language (Russian is taken care of). Please don’t contribute translated titles/descriptions without subtitles (people hate being lured into watching videos with titles in their native language only to find a video in English only.
And, b.t.w., I just finished a hellish first semester at uni here in Australia and now should have a bit more time to go for some ambitious YouTube projects. Fingers crossed.

update on epicycle animations by viewers:


some other related bits and pieces found by you:

[TheLuckySpades]

I'd just like to thank you for making videos like this.

Math videos like this are one of the reasons I stuck with math when it felt repetitive in school.
They are the reason I always knew that I knew nothing, but that I wanted to know more.
They inspired me to look into branches I would never have encountered on my own.

These types if video are my favorite, really entertaining and fun, rigorous enough that it can hold it's own without too much outside knowledge and intriguing enough to make me want to go out and learn more or in this case try to build the program myself. (Nothing "here's an interesting fact" type of video that also get really popular, it's a matter of taste).

Thank you for making these videos and keeping the quality of the math in it so wonderful.

[michalbotor]

you've outdone yourself sir. jesus. i still can't swallow the fact of how excellent in terms of insight this video really is.
you see, i've seen a fourier series before, but up until now i've never seen through it.
thank you hundredfold sir!
also, whoever makes you this awesome teeshirts is a freaking genius. every next one is at least as witty as the previous one.

[ReevansElectro]

I have been fascinated with the Fourier series for nearly 40 years but this was the first time that I saw it through the vision of epicycles! You brought great joy and wonderment to my life today. Thank-you.

[DeclanMBrennan]

A veritable epicyclic tour de force. Thank you.
A few years back, I was given a demo of a Victorian era mechanical machine, all brass and hardwood, for creating Fourier waveforms at the University of Manchester and now I have far more intuition for how it works.

[semicharmedkindofguy]

Your approach of showing the circles animations first and then explaining the circles using complex exponentials was very intuitive. Thank you!

[maydavidr]

Another superb popular math video from mathologer: visually appealing, whimsical and intellectually stimulating. What a splendid introduction to Fourier series. Your work is much appreciated!

[ShurikB]

Algebra autopilot,
That killed me.
Great job love the video.

[SpacePoolNoodle]

Dude the way you explained how Euler’s formula works was the best. I couldn’t have passed Signals and Systems without you.

[conrad]

Wow, somehow I have never completely understood why the inverse Fourier Transformation is calculated as it is .. thanks to this video now I know.

[bryanroland]

Loved it. Like 3Blue 1Brown, your videos make the "magic" and beauty of maths accessible to innumerate but curious viewers such as myself through the use of clever animations. And they don't get much cleverer than the ones in this episode.

[Aufenthalt]

This video should be mandatory in schools. Excellent as ever.

[pauselab]

that's probably the best fourier explanation on the site!

[antoninbesse]

What is wonderful about the mind is that it can grasp that something is truly awesome without understanding it. I am awed. Now I am inspired to try and understand.

[JAlexCarney]

I decided to watch youtube because I was stuck on my history of math project. (Simulating Plato's universe modle) turns out this video was exactly what I needed! Thanks Mathologer!

[dcterr]

Another wonderful video! Although I'd seen much of this elsewhere, I never made the connection between Fourier series and epicycles. Very inspiring!

[martinmartinmartin]

For years I had NO concept of how the Fourier Series "worked", but use it in total ignorance!
Thank you for the graphical explanations; suddenly I understand !

[bxdanny]

When I first heard about Fourier analysis, I thought the teacher was saying "Four-year analysis", that it was something you only learned in a four-year series of courses. Didn't make much sense, glad I was wrong.