Why did we forget this simple visual solution? (Lill's method) @Mathologer

Uploaded April 2019 | Updated September 2026, 2 weeks ago

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.

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:
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): 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 :)

Mathologer Patreon: patreon.com/mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)

$Why did we forget this simple visual solution? (Lills method) Todays video is about Lills 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, Horners form, synthetic division and a newly discovered incarnation of Pascals famous triangle. 00:00 Intro 04:14 Lills method 07:31 Free meal 09:51 Square turtles 11:39 Origami turtles 14:16 Iterative turtles 17:32 QED 24:00 Pascals turtle animation Here is the page with an implementation of Lills method for cubic polynomials that I show in the video. http://www.qedcat.com/misc/lill_method/ Its 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 Hulls webpage: http://mars.wne.edu/~thull/papers/amer.math.monthly.118.04.307-hull.pdf Lills 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 Kalmans book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lills 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 todays 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 :) Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)$

Heron’s formula: What is the hidden meaning of 1 + 2 + 3 = 1 x 2 x 3 ?

Euler’s Pi Prime Product and Riemann’s Zeta Function

$Pi is IRRATIONAL: animation of a gorgeous proof NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) This video is my best shot at animating and explaining my favourite proof that pi is irrational. It is due to the Swiss mathematician Johann Lambert who published it over 250 years ago. The original write-up by Lambert is 58 pages long and definitely not for the faint of heart (http://www.kuttaka.org/~JHL/L1768b.pdf). On the other hand, among all the proofs of the irrationality of pi, Lamberts proof is probably the most natural one, the one thats easiest to motivate and explain, and one thats ideally suited for the sort of animations that I do. Anyway its been an absolute killer to put this video together and overall this is probably the most ambitious topic Ive tackled so far. I really hope that a lot of you will get something out of it. If you do please let me know :) Also, as usual, please consider contributing subtitles in your native language (English and Russian are under control, but everything else goes). One of the best short versions of Lamberts proof is contained in the book Autour du nombre pi by Jean-Pierre Lafon and Pierre Eymard. In particular, in it the authors calculate an explicit formula for the n-th partial fraction of Lamberts tan x formula; here is a scan with some highlighting by me: http://www.qedcat.com/misc/chopped.png Have a close look and youll see that as n goes to infinity all the highlighted terms approach 1. Whats left are the Maclaurin series for sin x on top and that for cos x at the bottom and this then goes a long way towards showing that those partial fractions really tend to tan x. There is a good summary of other proofs for the irrationality of pi on this wiki page: https://en.wikipedia.org/wiki/Proof_that_π_is_irrational Todays main t-shirt I got from from Zazzle: https://www.zazzle.com.au/25_dec_31_oct_t_shirt-235809979886007646 (there are lots of places that sell HO cubed t-shirts) lf you liked this video maybe also consider checking out some of my other videos on irrational and transcendental numbers and on continued fractions and other infinite expressions. The video on continued fractions that I refer to in this video is my video on the most irrational number: https://youtu.be/CaasbfdJdJg Special thanks to my friend Marty Ross for lots of feedback on the slideshow and some good-humoured heckling while we were recording the video. Thank you also to Danil Dimitriev for his ongoing Russian support of this channel. Merry Christmas!$

500 years of NOT teaching THE CUBIC FORMULA. What is it they think you cant handle?

2000 years unsolved: Why is doubling cubes and squaring circles impossible?

$The Pigeon Hole Principle: 7 gorgeous proofs Lets say there are more pigeons than pigeon holes. Then, if all the pigeons are in the holes, at least one of the holes must house at least two of the pigeons. Completely obvious. However, this unassuming pigeon hole principle strikes all over mathematics and yields some really surprising, deep and beautiful results. In this video I present my favourite seven applications of the pigeon hole principle. Starting with a classic, the puzzle of hairy twins, we then have a problem with pigeons on a sphere, a pigeon powered explanation of recurring decimals, some party maths, a very twisty property of the Rubik’s cube, a puzzler from the 1972 International Mathematical Olympiad, and, finally, what some people consider to be the best mathematical card trick of all time. 00:00 Intro 01:49 Chapter 1: Hairy twins 06:46 Chapter 2: Five pigeons on a sphere 08:16 Chapter 3: Repeating decimals 13:14 Chapter 4: Partying pigeons 17:00 Chapter 5: Repeating Rubik 22:20 Chapter 6: Pigeons at the Olympiad 26:18 Chapter 7: The best mathematical card trick ever 31:24 Supporters Here are some links for you to explore. A scanned copy of Récréation mathématique: Composée de plusieurs problèmes plaisants et ... by Jean Leurechon on Google books. For the hair puzzle check out page 130) https://tinyurl.com/3b6amaxk The Pigeonhole Principle, Two Centuries Before Dirichlet by Albrecht Heeffer and Benoit Rittaud A very nice article about the origins of the pigeon principle and the hairy twins problem. Also features an English translation of the relevant page in Récréation mathématique https://tinyurl.com/hpkcuepx The 4/5 pigeons in a hemisphere puzzle was problem A2 of the 63rd Putnam competition in 2002 https://prase.cz/kalva/putnam/psoln/psol022.html Why are repeated decimals fractions? Watch this video on why 9.999... =10 for a big hint https://youtu.be/SDtFBSjNmm0 Or just skip straight to the answer https://en.wikipedia.org/wiki/Decimal_representation#Conversion_to_fraction If you dont own a Rubiks cube you can use this simulator to test what happens when you repeat some algorithms (move the faces using the keyboard) https://ruwix.com/online-puzzle-simulators/ The website of the International Mathematical Olympiad. https://www.imo-official.org . The problem I am considering in this video is Problem 1 of the 1972 olympiad. You can download all the problems from here. https://www.imo-official.org/problems.aspx Check out this very nice article about the Fitch Cheney five-card trick by Colm Mulcahy https://tinyurl.com/wttkfdwe Todays music is English Country Garden (and as usual Morning Mandolin at the end) from the free YouTube music library. Todays t-shirt I got from here https://www.theshirtlist.com/pizzibonacci-t-shirt/ Enjoy! Burkard$