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

2024-08-03

[public] 33.9K views, 4.73K likes, dislikes audio only

Today's mission: saving another incredible discovery from falling into oblivion: Steinbach's amazing infinite family of counterparts of the golden ratio discovered around 1995. 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://tinyurl.com/4jhju7dw

A paper citing Peter Steinbach's paper

https://tinyurl.com/48vcmfdn by Scott Vorthmann, David Hall, and David Richter. Vorthmann also wrote the software vZome, which is an emulator of the Zometool construction system. The original Zometool is based on phi. However Vorthmann also added a special mode in vZome based on the heptagonal field. See also this Jupiter notebook https://tinyurl.com/bduzayr3

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: /youtube/video/ubHVK71F01M

Phi and the TRIBONACCI monster

/youtube/video/e7SnRPubg-g

The fabulous Fibonacci flower formula

/youtube/video/_GkxCIW46to

Infinite fractions and the most irrational number

/youtube/video/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

I am collecting a whole pile of other interesting bits and pieces that did not get mentioned in the video and/or popped up in the comments in my post pinned to the top of the comment section of this video.

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?

5. Is it a coincidence that there is a 1/7 in my Binet's formula for the heptagon? (You can make the 1/5 th appear in Binet's formula itself by multiplying both denominator and numerator by phi +1/phi.)

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