Tuesday, March 12, 2013

Langton's Ant Trapped In a Circle, Triangle, or Near a Line Segment

  This Scratch project is a series of experiments using Langton’s Ant as the ‘lab rat’. 
  Langton’s ant is a cellular automaton invented by researcher Chris Langton in the early 1980s. The ant, once set in motion, crates an iconic picture of order emerging from chaos and is often a student’s first exposure to the study of deterministic chaos and nonlinear dynamics (chaos theory).
   When we observe complex behavior we tend to believe its behavior is the result of a complex set of rules. In the picture above, the path of a single Langton ant is shown. The ant started in the center of the screen and has worked its way to near the lower right corner of the screen. Out of the mess seen in the center, after over 10,000 applications of the rules governing its behavior, the ant starts building the diagonal ‘highway’! 
  I purposely stopped the ant before it built the highway to the boundary. What happen after the ant hits the side? Even though the rules that actually governs the behavior (movement) of the ant are quite simple, there is no way to predict what happens when the ant strikes the side of the rectangle. The actual path created by the ant has to be computed, step by step, in order to be seen. 
  Here are the rules:
   The ant is sitting on an infinitely large checkerboard made up of only white squares. The ant moves one square forward and, if the square is white, it paints it black, turns left 90º and moves one square forward. If the square is black, the ant paints it white, turns right 90º and moves one square forward. In either case, after painting the square white or black, the ant applies the same rule again. That’s it!. That simple rule creates the ant’s complex behavior.
   The following screen shot displays the ant wandering around the center of the screen and then building the highway.

   If you would like to find out what happen when Langton’s ant is placed inside a circle, square, or triangle go to the following link, download the project, and find out!
   The line segment at the bottom of the screen can also be picked up and placed closer to the ant's starting point. Below is a screen shot showing the ant building a highway after interacting with the line segment.

   Remember, there is no known method for predicting the behavior of the ant. Have fun experimenting with Langton's Ant!

Sunday, March 3, 2013

Mathematical Wallpaper

   The background for this project can be found in the book, The Turing Omnibus - 61 Excursions in Computer Science, by A. K. Dewdney, Chapter 1, page 3, and in Computer Recreations, A. K. Dewdney, Scientific American, September 1986. 

   I teach a Fractals and Chaos course for secondary math and science teachers as part of a graduate degree program. I assign Wallpaper as a programming exercise prior to assigning the Mandelbrot set algorithm as a programming exercise. An orderly method for exploring a section of the coordinate plane is needed in both.
   Wallpaper is an interesting application of the circle formula and odd and even numbers.
    Each set of unique inputs generates a unique pattern that is unpredictable. Even though we know the equation of a circle and how to test for even or odd, we have no way of predicting the pattern generated by the three inputs. it has to be computed to be seen.
Watch this short video to see Wallpaper in action.

   Go to this link to download the Scratch file for this project.
   I also have a free PDF file that goes into more detail about the math and the Scratch coding. It is available free, on request, by sending an email to grandadscience@gmail.com.
   Emails are never shared and are only used to transmit files.