Iterative Fractals Exploration

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The simplest fractals are constructed by iteration. This means that we apply a certain process repeatedly. For example, start with a filled-in triangle and remove the middle fourth. Repeat this process:

Sierpinski1.png Sierpinski2.png

Here we see respectively 1 and 2 iterations of this recursive process.

For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket. The gasket is self-similar. That is, it is made up of smaller copies of itself. "Big gaskets are made of little gaskets, The bits into which we slice 'em. And little gaskets are made of lesser gaskets And so ad infinitum."

  1. Iterate the process four times, shade in the resulting triangles: Sierpinski3.png
  2. Go to the Geometer’s Sketchpad. Go to the folder “Samples”, then “Sketches”, then “Geometry”. Open “Fractal Gallery”. Explore the Sierpinski Gadget, the Koch Snowflake and the Golden Spiral.
    a. Explain how the Koch Snowflake fractal is produced. (give a sketch.)
    b. Explain how the Golden Spiral is produced. (give a sketch.)