Visualising irrationality with triangular squares


Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers.

Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc.

What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1.

Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies.
https://arxiv.org/abs/0909.4913​

Footnotes:
-our nearest miss solutions like, for example,
15^2+15^2+15^2=26^2-1
correspond to the solutions of the equation y^2 – n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pell’s equation, which happens to have solutions for all integers n that are not squares.
-there is also a second type of nearest miss solutions like
4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses.
-the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution.
-The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots.
-lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop 🙂

Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support.

Today’s t-shirt is the amazing square root t-shirt (google “square root tshirt”). Note that the tree looks like a square root sign AND that the roots of the tree are really square.

Enjoy!

About agogo22

Director of Manchester School of Samba at http://www.sambaman.org.uk
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