Divine high PHI: The power of AB=A+B (Mathologer masterclass)

Today's mission: saving another incredible discovery from falling into oblivion: Steinbach's amazing infinite family of counterparts of the golden ratio discovered around 1990. Lot's of my own little discoveries in this one :)

00:00 Intro
05:53 Ptolemy
09:18 Perfect cut
16:01 Golden rectangle
22:03 Fibonacci
33:07 A+B=AB
45:48 Images and music
47:27 Thank you!


The slide show for this video is made up of a new record of 750 slides!

Peter Steinbach's papers articles:
Sections beyond golden:
https://archive.bridgesmathart.org/2000/bridges2000-35.html
Golden fields: a case for the heptagon:
https://www.jstor.org/stable/2691048

A good online writeup with some extra insights (on a site dedicated to sacred geometry!:
https://www.sacred-geometry.es/?q=en/content/golden-trisection

Alan H. Schoen's incredible infinite tiling site. For anybody who wants to explore some heptagonal Penrose rhombus tiling counterparts.

Also, check out the very good wiki pages dedicated to the golden ratio and the Fibonacci numbers.

The wiki page on Ptolemy's theorem features a great visual animated proof https://en.wikipedia.org/wiki/Ptolemy

Some relevant Mathologer videos:
The golden ratio spiral: visual infinite descent: https://youtu.be/ubHVK71F01M
Phi and the TRIBONACCI monster
https://youtu.be/e7SnRPubg-g
The fabulous Fibonacci flower formula
https://youtu.be/_GkxCIW46to
Infinite fractions and the most irrational number
https://youtu.be/CaasbfdJdJg

Golden ratio fact and fiction. Check out this paper by Georg Markowsky:
https://www.goldennumber.net/wp-content/uploads/George-Markowsky-Golden-Ratio-Misconceptions-MAA.pdf

For more on this also check out the book: The golden ratio by Mario Livio

Some questions for you to while your time away.
1. In the 3D golden spiral the left-over golden boxes converge to a point on one of the edges of the golden box we start with. In what ratio does this point divide the edge?
2. Which points in a golden rectangle can you reach by cutting off infinitely many squares as in the golden spiral construction? How about in 3d?
3. Nut out some details for the nonagon. What's Binet's formula in that case?
4. For which complex numbers n does Binet's formula spit out an integer/a real number?

e-mail from Marty: When I was doing my PhD, I read a paper by my supervisor, Brian White. It was on “surfaces mod 4”. So, it was a way to make surfaces multiple valued, turned the world of them into a ring, and mod 4 meant the obvious thing: four copies of a surface summed to 0. Anyway, I read the paper, and was trying to figure out why his theorem worked specifically for mod 4. And I asked: “Does it boil down to 2 x 2 = 2 + 2”? “Yep, that’s it." https://link.springer.com/article/10.1007/BF01403190

T-shirt: Fibonightmare
https://www.teepublic.com/t-shirts?query=Fibonightmare

Music: Kashido - When you go out and Ardie Son - Spread your wings

Enjoy!

Burkard