Times Tables, Mandelbrot and the Heart of Mathematics

Mathologer explains how modular multiplication tables mapped onto a circle generate cardioids, nephroids, and connections to the Mandelbrot set.

NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) The good old times tables lead a very exciting secret life involving the infamous Mandelbrot set, the ubiquitous cardioid and a myriad of hidden beautiful patterns. Time for the Mathologer to go on a serious fact-finding mission. For those of you who’d like to play around a bit with the stunning times table diagrams that we discuss in this video, download the .cdf file qedcat.com/cardioid.cdf and open it with the free cdf player which you can download from Wolfram Research (the people behind Wolfram Alpha and Mathematica). If you have access to Mathematica you can also open my .cdf file in Mathematica and play with the code. For those of you who are looking for a bit of a challenge, ponder this: 1) Starting with the fact that the nephroid arises from parallel rays being reflected inside a cylindrical coffee cup, try to convince yourself that the 3 times table really does produce the nephroid (some really neat geometry at work here, very similar to the argument for the cardioid that I talk about at the end of the video). (Added 8 November 2015 check out the proof at qedcat.com/nephroid_proof.pdf ) 2) Why do the diagrams for all the times tables have a horizontal mirror symmetry? 3) Try to explain the pretty patterns corresponding to the 51 and 99 times tables modulo 200 that I display in the video (around the 9:30 mark). 4) (For those of you with a very strong math background) Try to figure out why the cardioid shows up in the Mandelbrot set. The discovery of the stunning patterns that I discuss in this video is due to the mathematician Simon Plouffe. Check out this article tinyurl.com/o2hbtsa and his website plouffe.fr for other stunning visualisations using modular arithmetic. Quite a few animations have been contributed by various people and linked to in the comments: Here is one of the nicest ones by Mathias Lengler: mathiaslengler.github.io/TimesTableWebGL/ Enjoy! P.S.: The music we are playing at the end is called Shoulder Closure by Gunnar Olsen. It's part of the free YouTube music library. A really nice piece , isn't it?