Fractals

Iteration is the repeated application of the same rule. A fractal is a figure that is generated by iteration.

• Main topics

  • Chaos Game --> order, <<LINK TO PHOTOS>>
  • Cantor Set
  • Bifurcation daigram
  • Lorenz Attractor
  • Rossler Attractor
  • Dragon curve
  • Curlicue fractal
  • Koch Curve
  • Koch Snowflake
  • Sierpinski Triangle/Gasket - note Pascal Triangle mod 2 is Sierpinski Gasket
  • Apollonian Gasket - see this great Doodling in Math Class video by Viki Hart
  • random walks
  • Hilbert and Moore Space Filing Fractals
  • Wunderlich Space Filling Curve
  • Polya's Space-Filling Curve
  • Peano Space Filling Curve
  • Julia Sets
  • Mandelbrot Set
  • Sierpinski Carpet (flat)
  • Solid Sierpinksi Carpet -->


IFS - The Chaos Game - (Iterated Function Systems, recursion) - trees, ferns, ...
    1. initiator = starting picture
    2. generator = rule to repeat
    3. An virutal manipulative example - http://demonstrations.wolfram.com/TreeBender/

Key terms:
  1. Symmetry under magnification (1A, 1B)
  2. Scale invariance
  3. Self-similarity (“fractal properties”)
  4. How to make fractal – process is iteration
    1. initiator = starting picture
    2. generator = rule to repeat
  5. Famous fractals
    1. Sierpinksi triangle
    2. Koch Curve (1C-Kock Curve-everywhere continuous, no where differentiable)
    3. Cantor Set
  6. IFS – Iterated Function System – function is a rule
  7. Fractal Folds
  8. http://classes.yale.edu/fractals/Software/Software.html
  9. Random IFS / Edit -> Affince Transformations

• Dynamic Interactives and Manipulatives

  • Sierpinski Triangle/Gasket - Java style
  • Play Chaos Game.nbp - n-shaped figure, variy iterations, vary distances to vertex
  • ChaoticItineraryButRegularPattern.nbp - classic Chaos game on a triangle, 3 colors, ...
  • Explore NCTM's Fractal tool. Play with the Koch snowflake, a fractal tree, a reduced square, and the Sierpinksi triangle. As you progress through the stages, see what patterns you can observe in the images created and the table of values.
  • Fractal software list -

• Resources


• Photos (and short videos) from class

  • 3 days of fractals (December 2010) <<INSERT FLICKR LINKS HERE>>

• Videos