Ultimate Pythagoras: What is the hidden meaning of the twisted squares?

2022-10-15

[public] 13.1K views, 15.1K likes, dislikes audio only

A video on the iconic twisted squares diagram that many math(s) lovers have been familiar with since primary school. Surprisingly, there is a LOT more to this diagram than even expert mathematicians are aware of. And lots of this LOT is really really beautiful and important. A couple of things covered in this video include: Fermat's four squares theorem, Pythagoras for 60- and 120-degree triangles, the four bugs problem done using twisted squares and much more.

00:00 Intro

05:32 3 Squares: Fermat's four square theorem

12:51 Trithagoras

20:29 Hexagoras

22:06 Chop it up: More twisted square dissection proofs

23:42 Aha! Remarkable properties of right triangles with a twist

26:35 Mutants: Unusual applications of twisted squares

30:38 Op art: The four bugs problem

36:01 Final puzzle

36:32 Animation of Cauchy-Schwarz proof

37:16 Thanks!!

Here are a couple of links for you to explore:

My first Pythagoras video from four years ago:

/youtube/video/p-0SOWbzUYI

A collection of over 100 proofs of Pythagoras theorem at Cut-the-knot https://www.cut-the-knot.org/pythagoras/ (quite a few with animations) I cover proofs 3, 4, 5 ( :), 9, 10, 76, 104. Other proofs closely related to what I am doing in this video are 55, 89, and 116.

A very good book that touches on a lot of the material in this video by Claudi Alsina and Roger B. Nelsen - Icons of Mathematics: An Exploration of Twenty Key Images (2011). Check out in particular chapters 1-3 and chapter 8.3.

Fermat's four square theorem:

Alf van der Poorten's super nice proof https://arxiv.org/abs/0712.3850

Fibonacci seems to be the discoverer of the connection between Pythagorean triples and arithmetic sequences of squares of length 3 https://en.wikipedia.org/wiki/Congruum

Trithagoras:

Wayne Robert's pages. Start here and then navigate to "The theory to applied to the geometry of triangles" https://tinyurl.com/3k6afad4

M. Moran Cabre, Mathematics without words. College Mathematics Journal 34 (2003), p. 172.

Claudi Alsina and Roger B. Nelsen, College Mathematics Journal 41 (2010), p. 370. (Trithagoras for 30 and 150-degree triangles)

Nice writeup about how to make Eisenstein triples from Eisenstein integers

http://ime.math.arizona.edu/2007-08/0221_cuoco_handout2.pdf

More people should know about Eisenstein:

https://mathshistory.st-andrews.ac.uk/Biographies/Eisenstein/

Other twisted square dissection proofs:

There is an Easter Egg contained in the first proof. Five days after publishing the video only one person appears to have noticed it :) Here is an alternative version of the animation that only uses shifts that I put on Mathologer 2 https://www.youtube.com/shorts/vT9wUpu_Vco

The four bugs problem:

Actually I got something wrong here. Martin Gardner mentioned the four bugs for the first time in 1957 as a puzzle Martin Gardner actually mentioned the four bugs for the first time in 1957 as a puzzle (Gardner, M. November, 1957 Mathematical Games. Nine titillating puzzles, Sci. Am. 197, 140–146.) The 1965 article that is accompanied by the nice cover that I show in the video talks, among many other things, about the more general problem of placing bugs on the corners of a regular n-gon.

If you've got access to JSTOR, you can access all of Martin Gardner's articles through them.

https://www.jstor.org/journal/scieamer (all issues of the Scientific American)

https://www.jstor.org/stable/e24941962 (follow the Mathematical Games link)

https://www.jstor.org/stable/e24931930 (follow the Mathematical Games link)

https://www.jstor.org/journal/scieamer (all issues of the Scientific American)

Here are a couple of other online resources worth checking out.

https://tinyurl.com/4erz3zmf

https://tinyurl.com/ykvvj5sw

Explanation for distance 1: Because the bug that each bug is walking towards is always moving perpendicular to the first bug’s path, never getting closer or further away from the first bug’s motion. So it has to go exactly the same distance as it was at the beginning.

For the mathematics of various bits and pieces chasing each other check out Paul Nahin's book Chases and Escapes: The Mathematics of Pursuit and Evasion.

Solution for the puzzle at the end:

https://tinyurl.com/3eebxn2k

Today's music: A tender heart/The David Roy

T-shirt: google "Pythagoras and Einstein fighting over c squared t-shirt" for a couple of different versions.

Enjoy!

Burkard