NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) 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. The original Homer epicycle video “Ptolemy and Homer” by Ramiro Serra and Cristián Carman posted on their friend's Santiago Ginnobili's YouTube channel can be found here https://www.youtube.com/watch?v=QVuU2YCwHjw 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: https://mathematica.stackexchange.com/questions/171755/how-can-i-draw-a-homer-with-epicycloids Download my tweak of Anderstood’s Mathematica code as a Mathematica notebook here http://www.qedcat.com/misc/homer1.zip and in pdf format here http://www.qedcat.com/misc/homer1.pdf. The spirograph gif animation is from the Wiki page on spirographs https://en.wikipedia.org/wiki/Spirograph and is due to Michael Frey. A great interactive version Pierre Guilleminot of the square wave animation that I show at the end of this video lives here https://bl.ocks.org/jinroh/7524988 The GoldPlatedGoof video on Fourier analysis: https://www.youtube.com/watch?v=2hfoX51f6sg Today's t-shirt I got from here: https://www.teepublic.com/t-shirt/2066458-mathflix The music is "Mysteries" by Dan Lebowitz from the free YouTube music library: null As usual thank you very much to Danil and Marty for their help with this video. Enjoy!