The golden ratio spiral: visual infinite descent @Mathologer

Uploaded May 2018 | Updated September 2026, 2 weeks ago

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So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc.

Here are some articles that debunk a lot of the golden spiral in nature nonsense:
Clement Falbo
http://web.sonoma.edu/Math/faculty/falbo/cmj123-134

George Hart
georgehart.com/rp/replicator/replicator.html
youtube.com/channel/UCTl0dASnxto6j2wlVs5Bs2Q

Keith Devlin
http://devlinsangle.blogspot.com.au/2017/04/fibonacci-and-golden-ratio-madness.html

Here is a very good website devoted to everything to do with the golden ratio and Fibonacci
maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating).

Enjoy!

$The golden ratio spiral: visual infinite descent NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) So you all know the golden (ratio) spiral. But did you know that not only the golden ratio but really every number has such a spiral associated with it? And that this spiral provides key insights into the nature of a number. Featuring more proofs by contradiction by infinite descent (my current obsession), infinite continued fractions, etc. Here are some articles that debunk a lot of the golden spiral in nature nonsense: Clement Falbo http://web.sonoma.edu/Math/faculty/falbo/cmj123-134 George Hart http://www.georgehart.com/rp/replicator/replicator.html https://www.youtube.com/channel/UCTl0dASnxto6j2wlVs5Bs2Q Keith Devlin http://devlinsangle.blogspot.com.au/2017/04/fibonacci-and-golden-ratio-madness.html Here is a very good website devoted to everything to do with the golden ratio and Fibonacci http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html Thank you very much to Michael (editing), Marty (nitpicking) and Danil (translating). Enjoy!$

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$Visualising irrationality with triangular squares 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) 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 Pells 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. Todays 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!$

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