Iterative Fractals Exploration II

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Time-50.svg

Objective:

A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals.

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. 4 iterations of the Koch snowflake
  3. Next consider the Koch Snowflake. We can construct the Koch snowflake by the following process:
    • Draw an equilateral triangle
    • Divide each side in three equal parts
    • Draw an equilateral triangle on each side using the middle line segment as the base
    • Remove the middle line segment of the original side
    • Repeat!
    Construct your own Koch snowflake and go to at least 4 iterations.
  4. A tiling made up of two different sized Koch snowflakes
  5. Fractals and tilings: The image of the tiling by Koch snowflakes shows a combination of two main concepts in the course.
    • What iteration of the Koch snowflake is being used? Are the dark and light flakes showing the same level of iteration?
    • What is the symmetry group of the tessellation?
    • Can you think of some interesting tessellations we could construct using the Siepinski triangle? Draw at least 2.

Handin: A sheet with answers to all questions.