Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

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

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:

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 :)

Today's music: Antionetta by Boreís and Dark tranquility by Anno Domini Beats

Enjoy!

Burkard

[hydra]

24:22 For the 3, 4, 5 triangle the line connecting the incenter and the Feuerbach point is parallel to the shortest side. Thus, the parent triangle of the 3, 4, 5 triangle is the degenerate 0, 1, 1 triangle (and its clear why this construction cannot be taken further).

[royalninja]

Regarding the tree puzzle at 15:50: The bottom square's area is 1. The Pythagorean theorem states the two squares attached it share the same total area, so they are each 1/2. The total area so far is then 2, with 1 contributed from the big square and 1/2 + 1/2 = 1 contributed from the small squares. The next level down, the tinier squares attached to one of the small squares has to add up to 1/2, so they are each 1/4. There are 4 of them, so the total area of these squares is 1, and the total area is now 3. You continue down the line, adding 1 to the total area for each iteration of the tree. There are 5 iterations, so the total area is 5.

[aeschynanthus_sp]

My father was a carpenter. He built various buildings, and the last one was a cottage where I helped. We checked that the corner was at right angle using the 3-4-5 measurement.

[johnklinger]

One neat fact that was left out is that when 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.

[johnmeyers]

I've been fascinated with Fibonacci numbers and Pythagorean triples since I discovered them when I was about 8. 45 years later you taught me some new things and helped me understand the "why" behind some of what I already knew. Thank you.

[thephilosophyofhorror]

Despite serious competition, Mathologer remains the greatest math channel imo ^^ Thanks for another awesome video!

[contrawise]

I am not a studied mathematician by anyone's measure. Yet, I carry away so much from your videos! Thank you so much for your well-constructed presentations. This one was wonderfully startling!

[jakoblenke]

16:00
I found the area of the tree to be 5, since we have a depth of 5 and for every iteration, the new squares sum up to 1, thus a a tree with infinite iterations has an infinite surface area (still counting overlapping surfaces)

[richardfredlund]

11:56 challenge question: The Pythagorean triple (153, 104, 185) corresponding to the box
[ 9, 4, 17, 13 ]. If you call the children A, B, C, the 153, 104, 185 is the A'th Child of 'CCC'

[sixhundredandfive]

Whenever I'm finding myself lost or at a dead end with my own mathematical work you seemingly post a video that helps me along my path. Thank you.

[TigruArdavi]

Awesome, Mathologer! Another ab-so-lute-ly delightful journey. Apart from its important and most beneficial applications, Mathologer never ceases to amaze with another revelation on how maths has just this amazing beauty and harmony in itsself. This channel is such a gem on YT. ✨Thank you very much, indeed, Sir. 🙏

[axisjayy]

The beauty of the interconnectedness of mathematics

[danielhmorgan]

I was moved by this, almost to tears. what a great treatment of a rich subject. Thank you, thnkyu, thku, thx...

[denelson]

My favourite way to calculate primitive Pythagorean triples is to just use Euclid's formula that 3blue1brown showed us, using two coprime integers that are not both odd, and in fact, the two integers you need to run through Euclid's formula to get a specific primitive Pythagorean triple can be found in the right column of the Fibonacci box corresponding to said primitive Pythagorean triple, and this always works for any such box you choose.

[DeclanMBrennan]

This is amazing. Everything truly is connected to everything. I could hardly have been more surprised if Pascal's triangle had also made an appearance. 🙂

[BrandonWillWin]

I’m stunned. What a sublime concept, especially the animations that produce the cool little fractal trees

[iveharzing]

Every single video of yours has so many interesting mathematical connections!
I always get excited while watching them!

[sinecurve]

24:13 The location of the center of Feuerbach circle for the 3-4-5 triangle is on the same horizontal line as the incircle, therefore the outlined procedure will produce a degenerate triangle (a line).

[AnonimityAssured]

Mathologer videos are always such an inspiration to me. I'm no mathematician, but I enjoy a bit of mathematical dabbling. Most of my exploration is what might be called empirical mathematics. In short, I look for patterns without bothering too much about proofs. To test my pattern-finding ability, I paused the video at 29:01, to see whether I could identify the next few members of the family. I got the following:
9² + 40² = 41²; 11² + 60² = 61²; 13² = 84² = 85²; 15² + 112² = 113²; 17² + 144² = 145².
The general pattern could be expressed as (2n + 1)² + (2(n² + n))² = (2(n² + n) + 1)², where n is a natural number.
The pattern for the family shown at 30:12 was even easier to identify. The next few members were:
63² + 16² = 65²; 99² + 20² = 101²; 143² + 24² = 145²; 195² + 28² = 197².
The general pattern for this could be expressed as (4n² − 1)² + (4n)² = (4n² + 1)², where n is a natural number.

[lennyvoget]

The 153, 104, 185 triple at 12:00 is the box 9, 4, 13, 17 and you get there by navigating right right right and left :)