One of the many great things about my readers is how you folks keep me up to date with any new crap that springs up, so that I don’t need to spend so much time hunting down the real good stuff. There’s a beautiful piece of crap on youtube that was pointed out to me by one of you guys. It’s really a wonderful bit of circularity.

Circularity is something that I find beautiful in math. What I mean by circularity is that because numbers are closed, you can run around in circles playing games with that closure. Another post that I’ve got in progress is talking about RSA encryption, which is a beautiful example of circularity. You start with a message, encoded as a number, M. Then you take a particular set of three numbers, N, D, and E. If you raise M to the Dth power modulo N, you get a new number. M’. If you raise M’ to the Eth power modulo N, you get the original M. You’re never taking roots – but the two exponentiations cancel each other out modulo N. It’s beautiful, and astonishing, and yet it makes perfect sense.

That’s a complicated example of circularity. A simpler one, also involving modulo arithmetic, is to look at the tempered music scale. Let A=0, Bb=1, B=2, C=3, Db=4, D=5, Eb=6, E=7, F=8, Gb=9, G=10, Ab=11. Now, start at A, and follow through musical fifths – that is, go from A(0) to E(7). Then E(7) to E+7=14 mod 12 = 2 = B. Then B to Gb(9). Then Gb to Db(4). Then Db to Ab. Then Ab to Eb. Then Eb to Bb. Then Db to F. Then F to C. Then C to G. Then G to D. Then D to A. You’ve taken twelve steps of fifths, and wound up where you started. So by following through one of the natural musical elements of harmony, you’ve got a circle that visits each note exactly once. Looked at mathematically, it’s trivial. But it’s still pretty cool.

It’s pretty easy to trick yourself with circularity of you’re not careful. You can find what appear to be amazing numerical coincidences, because you don’t realize that you’ve created a circle.

The target of this posts isn’t an example of that. It’s a really trivial circle.

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