Tuesday, January 29, 2013

The Lorenz Attractor

    In 1962 Dr. Ed Lorenz, a meteorology professor at MIT,  published a paper in the Journal of the Atmospheric Sciences titled Deterministic Nonperiodic Flow. His paper quickly caught the attention of physicists and mathematicians alike and turned out to be the spark that caused the study of nonlinear, chaotic, systems, to explode into a new branch of science. His paper popularized the idea called the butterfly effect, that is, a butterfly flapping its wings in China would effect the weather halfway around the world.

    Professor Lorenz refined his initial set of 12 differential equations that modeled the simplest weather system possible, a convection cell, into a much smaller set of three such equations. The three equations are listed in the Project Notes section of the Scratch project.

   I've written a pdf file titled The Lorenz Attractor in Scratch that explains the physics as simply as I know how to do it and works through the Scratch scripts that plots the motion of the single point in phase space that represents the state of the convection cell at any point in time. The resulting image, called the Lorenz Attractor, has become the iconic image for the butterfly effect and chaos.


   You can view the Scratch project and/or download the sprites and scripts for the project by clicking on the flowing link.


   If you would like a free copy of The Lorenz Attractor in Scratch pdf send an email request to grandadscience@gmail.com (no spam, nothing for sale). I also have a pdf file of his now classic paper, Deterministic Nonperiodic Flow, that I can email to you on request. Accept the correctness of his mathematics and read the paper for the important ideas it contains.

Friday, January 18, 2013

A Pursuit Curve Problem

   As a math and science educator I subscribe to a number of professional periodicals. One of  my favorites is THE PHYSICS TEACHER. I was excited to find the following problem in the September 2012 issue.

Physics Challenge for Teachers and Students
Boris Korsunsky, Column Editor
West High School, Weston, MA 02493

A Futile Chase
Two turtles, A and B are relaxing at the water’s edge a distance d apart. Then A begins to swim away from the shore. B gives chase, taking off at the same moment. During the chase A keeps swimming directly away from the shore while B keeps swimming directly towards A. The speeds of both turtles are the same. Find the distance between A and B after a long time interval.           

   As soon as I read this "challenge" problem I quickly decided not to pursue a purely mathematical solution but, instead, model the problem in Scratch. I couldn't resist the  fun of making two sprites in the form of turtles and then animating both turtles according to the conditions stated in the problem!
   Here is a diagram of the starting position for both turtles.
   The chase is futile because turtle B will never catch turtle A. Why? Both turtles travel with the same velocity, they are not traveling towards each other, and they start the chase with a separation distance d. Still, the geometry of the situation guarantees that turtle B will get closer to turtle A. What's the closest turtle B gets to turtle A? In other words, what is d after a long time interval?

   I wrote the Scratch scripts, varied the starting distance d, looked at the separation distance when both turtles were traveling in the same direction, and formed a conjecture. Doing the math to prove my conjecture seems pointless to me because I so strongly believe my conjecture to be true!
   If you would like to form your own conjecture, here's the link to my Scratch program you can download and play with.
   This is a short video of the program in action.

Tuesday, January 15, 2013

Ant Poly - a Generalized Polygon Script

   Many years ago, when classroom computers first became available, we created Computer Labs at each of the three middle schools in our district. Every student cycled through a lab, for one period a day, for a semester. We wrote our own curriculum and two-thirds of the class time was spent teaching the kids Logo, the first kid-friendly programming language to originate at MIT. Logo, like Scratch, makes it very easy to turn the normally static study of geometry into a very dynamic experience for the kids.

   As middle school students, they were also butting heads against an algebra class and it wasn’t a pleasant experience for most of the kids. In the computer lab, we spent a lot of time working on the variable concept giving the kids problems and projects to program that required the use of one or more variables in the code.

   One of our programming goals was to have the kids develop a generalized polygon procedure (script). View this short, one-minute video to see the generalized polygon script in action.

     A lot of geometry content is used in building the script. For years we saw unusually high scores in the geometry section of the standardized math tests the students took every year. Alas, as I write this, every one of the labs is gone in that district.
    I've written a detailed explanation of how to build the generalized polygon script in Scratch. To obtain a free copy, email your request to grandadscience@gmail.com and I will email you the pdf file.
  Go to this link
 to download a Scratch program file that includes the ant, the two sliders, and the sand background ready for you to build the script as instructed in the above pdf file.