Fractals

toc =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, <>
 * 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
 * blog entry on Sierpinski Triangle with video links - []
 * 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 - []

Key terms:
 * 1) Symmetry under magnification (1A, 1B)
 * 2) Scale invariance
 * 3) Self-similarity (“fractal properties”)
 * 4) How to make fractal – process is iteration
 * 5) initiator = starting picture
 * 6) generator = rule to repeat
 * 7) Famous fractals
 * 8) Sierpinksi triangle
 * 9) Koch Curve (1C-Kock Curve-everywhere continuous, no where differentiable)
 * 10) Cantor Set
 * 11) IFS – Iterated Function System – function is a rule
 * 12) Fractal Folds
 * 13) @http://classes.yale.edu/fractals/Software/Software.html
 * 14) 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

 * Rice University - great site
 * Wrapped in Pride -
 * Substitution Tilings -
 * African Fractals - there is also a TED video -
 * Fractal Tiling -
 * Tessellations -

• Photos (and short videos) from class

 * 3 days of fractals (December 2010) <>

• Videos

 * Doodling in Math Class by Viki Hart, contains binary trees, Sierpinski triangle, ...
 * Dooodling in Math Class by Viki Hart, Apollonian gaskets
 * Everything is Fractal (IFS) -
 * IFS - Growth of a fractal tree -
 * Sierpinski triangle - infinite zoomin in - self-similarity
 * Sierpinski tetraahedron - rendered roatating in space
 * Sierpinksi tetrahedron - you can wrap your hands around it, it's dimension = 2 (wow!) with infinite surface area.. The fractal is incorrect: there should not be the inner group of tetrahedrons within the outer 4 sets of tetrahedrons.
 * Menger Sponger Zoom -