Times Tables, Mandelbrot and the Heart of Mathematics

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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 http://www.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 http://www.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 http://tinyurl.com/o2hbtsa and his website http://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:


Burkard Polster and Giuseppe Geracitano

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?


The True says:

These sets always felt like the key to dimensional theory, because of the way they behave and corelate. IE: A 3D object is viewed by our eyes as a 2D image. Really interesting, when I was a kid my Pal286 PC had a Mandelbrot generator that evolved fractals based on input numbers, which seemed to pinpoint a section of the Mandelbrot set and zoom in infinitely with chromatic variations that were beautiful. I wish I still had that program to show my children, since it made math fascinating for young me.

Faraday Sage says:

3:07 math made a butt and anus. Numbers truly are everything.

Jason Usborne says:

Witness this in growth of life, birth of Universe. Pattern of patterns!

taha amar 2 says:

Explanation is very useful and beautiful and entertaining and wonderful and above magnificence

Dakota Todd says:

I think its interesting that when he was breaking it down to doing 3× or 4× and getting the 2 or 3 point circle thing it's almost like its mimicking the perception of viewing say of dimensions like how seeing the 2nd dimension from a 3rd dimension point of view if that makes sense idk I could be rong on this but idk just off of the first perception of it I started thinking about that

AlphaMineron says:

Can someone point me in the right direction to actually simulating this in Python???

Harvey Harbicht says:

The more of your videos I watch the more my mind boggles at how everything is math and physics and how everything fits together.

Please do me a gigantic favor. Invent a time machine and go back to 1965 or so. Find little me and show me this stuff. Then tell me to keep my stupid ass in school and get a Mathematics or Physics Ph.D. PLEASE?!?! lol

That animation you did would be a perfect subject for a computerized laser show. Imagine those patterns moving across an entire room.

Aditya S says:

Hey…if we take pi times table…or e times table…

Sourav Chanda says:

can we get this functionality which u r making in a exe file so we too can make it?? ur response will be helpful

Denis Black says:

4:40 that looks weird lol 😀

Jorge Naiene says:

Maths is not lame. At all; Interesting… I like stuff like this. HEY THANX MR. MATHOLOGER

Carlos Augusto says:

Fourier series?

Dennis Maine says:

thank you for sharing this beautiful work!

Waggy Waagg says:

This video is a proof that our world is a computer reality

Arena Pixel says:

Se parece a un culo

Mr. Cub Fan 415 says:

3:24 Horizontal symmetry because it’s a 2x table 🙂

Also, the multiplier seems to always be 1 greater than the number of axes of symmetry of the diagram (for integer multipliers only, of course). 2x table has 1 axis of symmetry, 3x table has 2 axes of symmetry, etc.

milton jeong says:

1÷99=0.010101,2÷99=0.020202,3÷99=0.030303……97÷99=0.979797,98÷99=0.989898…watching your video, I discovered that. 99 made squares..so i wondered about 99.

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