Why don’t they teach this simple visual solution? (Lill’s method)

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From Mathologer.

Today’s video is about Lill’s method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horner’s form, synthetic division and a newly discovered incarnation of Pascal’s famous triangle.

00:00 Intro
04:14 Lill’s method
07:31 Free meal
09:51 Square turtles
11:39 Origami turtles
14:16 Iterative turtles
17:32 QED
24:00 Pascal’s turtle animation

Here is the page with an implementation of Lill’s method for cubic polynomials that I show in the video.

http://www.qedcat.com/misc/lill_method/

It’s an adaptation of this webpage

http://heim.ifi.uio.no/magho/lill/

(I have not been able to find out who put this together originally).

The article that inspired this video is this:

Thomas C. Hull, Solving Cubics With Creases: The Work of Beloch and Lill, The American Mathematical Monthly , Vol. 118, No. 4 (April 2011), pp. 307-315. Here is a link to this article on Thomas Hull’s webpage: http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf

Lill’s original paper:
http://www.numdam.org/article/NAM_1867_2_6__359_0.pdf

Other good references include:
Polynomials as polygons by Serge Tabachnikov
https://www.math.psu.edu/tabachni/prints/Polynomials.pdf

Dan Kalman’s book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lill’s method)
https://books.google.com.au/books?id=JPq0pS3wrx4C&pg=PA7&source=gbs_toc_r&cad=3#v=onepage&q&f=false

Thank you very much to Marty, Karl and Danil for their help with this video.

One version of today’s math t-shirt (Zombie addition): https://www.redbubble.com/people/manikx/works/8929883-zombie-math?p=t-shirt

The piece of music at the end is called "Fresh fallen snow" by Chris Haugen from the free YouTube music library.

Really neat 1-line Mathematica code for the generation of the Pascal turtle which appeared on Reddit after the video was posted there:
Graphics[Table[Line[ReIm[Accumulate[Table[2^(-n/2)Binomial[n,k]Exp[I(4+2k-n)Pi/4],{k,-1,n}]]]],{n,0,7}]]
and another nice implementation in Python (with a real turtle graphics turtle) by Alex Hall https://repl.it/repls/DeepskyblueFractalPoint

Enjoy 🙂

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