# 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)

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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