Showing posts with label Koch. Show all posts
Showing posts with label Koch. Show all posts

Sunday, June 23, 2013

Preparing to Teach a Fractals and Chaos Course

   I will soon be teaching (for the fifteenth time) an introductory course called Fractals and Chaos. The students are middle and secondary school math and science teachers. As part of the course, they are required to do a certain amount of programming and I have been converting the program assignments from another language to Scratch. Below are just three of the conversions that have been uploaded to My Stuff on Scratch. 
   The first two projects are examples of what are called strange attractors. The first really famous strange attractor was the Lorenz attractor (already in My Stuff).  Before attempting to understand strange attractors, we study the concept of a mathematical attractor using a simple linear equation and iteration.

The Hénon Attractor

The Rössler Attractor

   One of the curves that forced mathematicians to redefine the definition of 'curve' was the Koch snowflake. The snowflake curve is an example of a similarity fractal. It's mathematical properties are counter-intuitive. For example, it's perimeter is infinite in length but it bounds a finite area. Students are assigned the task of constructing their own similarity fractal.
The Koch Snowflake

   Additional information about each of these projects (in the form of PDF files) can be had by sending an email request to

Thursday, November 29, 2012


   The purpose of this blog is to provide detailed information about the mathematics, science, and programming techniques embedded in the Scratch programming projects I so love doing.
   I was motivated to start this blog by a comment one viewer made in regard to a project I posted on my Scratch page at
   Here is a screen shot of the project. The squareflake has the unusual property of always maintaining the same area as its originating square (in yellow) even if the perimeter is increased indefinitely by repeatedly applying the Lindenmayer rule!

   The aforementioned Scratch user had viewed my Lindenmayer System Squareflake project at 
and posted this comment,

AWESOME! How does it work?

   How code works is always a good question. I did include numerous comments in the Scratch scripts included in the project but they would most likely be helpful only to those that already knew something about Lindenmayer systems.
   Lindenmayer systems are not difficult to understand.  In fact, I did the project because I did not understand a Lindenmayer system!
   For those with an interest in learning the mathematics behind the code, I’ve written a detailed description of what Lindenmayer systems are, how they work, the applied mathematics, and how to program such systems in Scratch.

   To obtain a free pdf file for the Lindenmayer project simply email a request to and ask for Lindenmayer Systems in Scratch.

   Registered Scratch users can download the project by clicking on the above link. To download Scratch for the PC and Mac and become a registered user go to