Channel: Mathologer

Category: Education

Tags: lill's methodruler and compasseduard lillcarlyle circlehorner formorigami mathematicsthomas hullmargarita piazzola belochsynthetic division

Description: 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 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: 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 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) 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 repl.it/repls/DeepskyblueFractalPoint Enjoy :) Mathologer Patreon: patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)