Showing posts with label Sierpinski. Show all posts
Showing posts with label Sierpinski. Show all posts

Tuesday, October 8, 2013

Five More Methods for Computing the Sierpinski Triangle

   In a previous post (see An Iconic Image of Deterministic ChaosFebruary 23, 2013), I shared a Scratch project that used mathematician Michael Barnsley’s Collage theorem to compute the Sierpinski triangle.
   I know five other methods for computing the same image;
(1) by playing the Chaos game, 
(2) by coloring the odd numbers in Pascal’s triangle, 
(3) using recursion and the initiator-generator method, 
(4) using the Lindenmayer L-system method, and 
(5) by using Wolfram’s linear cellular automaton, Rule 90.
   It’s the last method, Wolfram’s Rule 90, that is the subject of this post. Here is Rule 90.

   For any three-cell group, the state of the center cell, in the next generation, is determined by the black and white pattern of the three cells. Examine Rule 90 and you will find that the center cell transforms to a black cell in the next generation if the XOR operator (black = 1 or white = 0) of the left and right cells returns a 1 (black).
   You can view this project in action by clicking on this link. You do not need to have Scratch installed as the program will run in your browser.

   If you would like a detailed walk-through of Wolfram’s Rule 90, in a PDF file, email your request to:

Saturday, February 23, 2013

An Iconic Image of Deterministic Chaos

   The Sierpinski triangle, along with the Mandelbrot set, are two of the iconic images of Fractals and Chaos theory. 
   The mathematician Michael Barnsley created a theorem called the Collage Theorem that provides the mathematics for playing the 'Chaos Game' (the common source for the Sierpinski Triangle) that will compute ANY image! 
   Perhaps you've seen the ferns (a soon to be uploaded Scratch project) and other images produced by the Collage Theorem (called Iterated Functions Systems by Barnsley).
   Geometrically, the Collage Theorem is the mathematics for creating an affine geometric transformation which is a combination of translations, rotations, and scalings (taught in secondary mathematics). Every image has a unique set of values for the coefficients a, b, c, d, e, and f of the transformation equations. Plug those values into an algorithm like the one in this project and compute the image! 
   Here is a very short video of the Scratch project.
   The Chaos Game is usually presented as a random application of the midpoint theorem, a typical Algebra I topic. In the future I will upload the midpoint formula project as used in playing the Chaos Game. It computes the same image computed in this project.
  Click on this link to go to the Scratch web site and download the project.
   I have also written a PDF document that contains biographical information about Dr. Barnsley and works through the mathematics used in the project. Send an email to
to request a free copy.