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## Fascinating Fractal Explorations

Several weeks ago, the notion of spirolaterals was introduced, including the use of a great on-line spirolateral generation machine. The software was produced by a NSF project Pattern Exploration, that focused on integrating math and science in the middle grades....but the math understandings involved seem more appropriate for the high school levels.

In addition to the Spirolateral program, the Pattern Exploration web site offers fascinating fractal-based programs for students to explore. Instructions are available, but it seems fine to just jump in, play, and discover what is happening. However, the important next step is to reflect on the visual patterns formed and try to understand the underlying mathematics or geometry.

The set of programs includes:

• Pythagorean Trees starts with a triangle on top of a rectangle as a template, then generates iterated images. Each iteration attaches the base of a rescaled template to the open sides of the previous triangles.
• Chaos Lab offers five dynamical systems programs: Time Series, Graphical Iteration, Frequency, Bifurcation, and Time Series 2. If you have read much about fractals and chaos, you should recognize the iteration and bifurcation images.
• Coastlines uses mathematical ideas to generate coastlines. The underlying idea is iteratively dividing of segments at a specific point and displacing them in a direction. Using color options, you can get close to images of actual coastlines.
• Paperfolding at first seems misnamed, until you have played with it for a while. The images are produced by iterating controlled folding patterns...and even unfolding at specific angles. Unusual!
• The Chaos Game is the standard model with some variations to explore. A moving game point is located relative to the three vertices of an equilateral triangle. The game rules: Roll a 3-sided die (corresponding to 3 vertices) and move the current game point half-way towards the vertex determined by the die. If you have never seen this game, play it and be surprised!
• Sierpinski Hunt is a game version of the Chaos Game...something sounds redundant?
• Plus three different variations of the Sierpinski family, via the Chaos Game. Again, you should recognize some of the visual images being generated...and it will test your abilities to understand all of the underlying geometry and mathematics.
Have fun exploring!