Math Circles on Fractals

Winter 2005

University of Waterloo - Waterloo, Ontario, Canada

Instructor

Lectures

Homework and Test

1. Homework exercises you might try for the first week.
2. Homework exercises for the second week.
3. Homework exercises for the third week.
4. Test at the end.

Fractal References

• LECTURE 1 - The Koch curve and Sierpinski triangle; fractals and non-fractals
  • Paul Bourke Fractals and Chaos pages containing many examples of fractals.
  • Self Similarity of a coastline.
  • Fractals in Wikipedia; the free web encyclopedia.
• LECTURE 2 - Fractal dimension and the Chaos game
  • The Cantor Set
  • The Cantor Set and Menger Sponge [Paul Bourke]
  • Description of the Chaos Game
  • Chaos Game for the Sierpinski triangle [Shodor].
  • Chaos Game for the Sierpinski triangle [Univ Bremen].
  • Relatives of Sierpinski - Game [Univ Bremen].
  • Chaos Game using a graphical Iterated Function System editor.
  • Fractal trees, etc.
  • Fractal coastline
  • Fractal landscapes
• LECTURE 3 - Iterating functions; complex numbers
  • Nonlinear Web applet [Boston University] displays cobweb diagrams for the quadratic and cosine maps.
  • Iteration of the Logistic Map [Andy Burbanks]
  • Chaos Lab [University of Bremen] displays cobweb diagrams for the logistic map.
  • An introduction to complex numbers
  • Dave's Short course on complex numbers
• LECTURE 4 - Julia and Mandelbrot sets
  • Filled-in Julia set applet [Shodor]
  • Filled-in Julia set applet (Beware: the positive imaginary axis points downwards; for instructions see their Mandelbrot set) [tzingaro]
  • Mandelbrot and filled-in Julia sets [Boston University]
  • Mandelbrot and Julia sets [Mark McClure]
  • The Julia set trip [Evgeny Demidov]
  • XaoS real-time interactive fractal zoomer for Windows and Mac OS 9 (but not Mac OS X)
    [Thomas Marsh and Jan Hubicka] The tutorials are good. Read:
    • Help->Tutorials->Math behind fractals->Whole story
    • Help->Tutorials->An introduction to fractals->Mandelbrot set
    • Help->Tutorials->An introduction to fractals->Julia set
• OTHER LINKS
  • A Fractals Unit for Elementary and Middle School Students by Cynthia Lanius.
  • The Fract-ED Information Pages, an introductory fractal tutorial for high school or college entry-level students.
• LOTS MORE REFERENCES AND LINKS from a university course I gave.