Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

From Mathologer.

In 2007 a simple beautiful connection Pythagorean triples and the Fibonacci sequence was discovered. This video is about popularising this connection which previously went largely unnoticed.

00:00 Intro
07:07 Pythagorean triple tree
13:44 Pythagoras’s other tree
16:02 Feuerbach miracle
24:28 Life lesson
26:10 The families of Plato, Fermat and Pythagoras
30:45 Euclid’s Elements and some proofs
37:57 Fibonacci numbers are special
40:38 Eugen Jost’s spiral
41:20 Thank you!!!
42:27 Solution to my pearl necklace puzzle

The two preprints by H. Lee Price and and Frank R. Bernhart and another related paper by the same authors:

The wiki page on Pythagorean triples is very good and very comprehensive

Wiki page on Pythagorean triple trees

Mathematica code for the Pythagorean Christmas tree by chyanog

Geoalgebra app for the Pythagoras tree fractal by Juan Carlos Ponce Campuzano

Connection to the Farey tree/Stern-Brocot tree in a paper by Shin-ichi Katayama

David Pagni (on the extra special feature of the Fibonacci number)

Eugen Jost’s Fibonacci meets Pythagoras spiral (in German)

Bug report:
06:06 – right circle doesn’t touch line (I mucked up 🙁

Puzzle time codes:
11:41 Puzzle 1: a) Fibonacci box of 153, 104, 185 b) path from from 3, 4, 5, to this triple in the tree
16:02 Puzzle 2: Area of gen 5 Pythagorean tree
25:55 Puzzle 3: Necklace puzzle

Some interesting tidbits:
Jakob Lenke put together an app that finds the route from 3,4,5 to your primitive Pythagorean triple of choice inside the tree. Thanks Jacob

theoriginalstoney and Michael Morad observed that at 39:28 (last section, extra special Fibonacci) the difference between the two righthand numbers (4 and 5, 12 and 13, 30 and 34, 80 and 89) are also squares of the Fibonacci numbers: F_(2n+3) – 2 F_(n+1) F_(n+2)=(F_n)^2

Éric Bischoff comments that the trick to get a right angle at 25:40 is popularized in French under the name "corde d’arpenteur". This term refers to a circular rope with 12 equally spaced nodes. If you pull 3, 4 and 5-node sides so the rope is tense, you get a right angle. See article "Corde à nœuds" on Wikipedia

Various viewers told me what F.J.M. stands for: Fredericus Johannes Maria Barning, Freek, b. Amsterdam 03.10.1924, master’s degree in mathematics Amsterdam GU 1954|a|, employee Mathematical Center (1954-), deputy director Mathematical Center, later Center for Mathematics and Informatics (1972-1988) Deceased. Amstelveen 27.06.2012, begr. Amsterdam (RK Bpl. Buitenveldert) 04.07.2012.

John Klinger remarks that if the four numbers in the box are viewed as fractions, the two fractions are equal to the tangents of half of each of the two acute angles of the triangle.

Colin Pountney: Here is another piece in the jigsaw. The link to Pascals triangle. It only works for the Fermat series of triples (ie the set of "middle children"). Choose any row in Pascals triangle. Multiply the odd entries by 1, 2, 4, 8, ….. and add to get the top left entry in a Fibonacci box. Do the same with the even entries to get the top right entry. For example taking the 1 5 10 10 5 1 row, we have top left number = 1*1 + 2*10 + 4*5 = 21. Top right number = 1*5 + 2*10 + 4*1 =29. For example taking the 1 6 15 20 15 6 1 row we have top left = 1*1 + 2*15 + 4*15 + 8*1 = 99. Top right = 1*6 + 2*20 + 4*6 =70. Not obviously useful, but it seems to make things more complete.

Ricardo Guzman: Another cool property of Fibonacci numbers: Take any 3 consecutive Fibonacci numbers: 55,89,144. The difference of squares of the larger two, divided by the smallest, is the next Fibonacci. …. Thus, in interesting ways the Fibonacci numbers are intertwined with the squares.

CM63: This suggested the attached figure to me.
In reply I suggested to extend this picture a spiral using these identities: phi^2=phi+1, phi^3=phi^2+phi, phi^4=phi^3+phi^2 🙂

According to this note on the relevant wiki page if you take overlaps of the Pythagorean tree into consideration the area of the tree is finite.

Today’s music: Antionetta by Boreís and Dark tranquility by Anno Domini Beats
Today’s t-shirt: google "Fibonacci cat t-shirt" for a couple of different versions. I just bought this t-shirt from somewhere but I think the cat is supposed to be superimposed onto this type of Fibonacci spiral