CS142: Topics for the Week of April 9, 2007

• Thursday: Review last lab: Lab 9.

• New lab: Lab 10 Fractals.

• Tribe Helps Linguist Argue with Prevailing Theory

• Modifying Images: Just added to Jar file (TextureCode.jar)- Wave filter

• Lorentz attractor and povray code: lorenz.pov
• Recursive tree: and povray code: tree.pov

Fractals

• What is a fractal?

  • What is a fractal?": Mandelbrot derived the term "fractal" from the Latin verb frangere, meaning to break or fragment. Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of "worlds within worlds"...

  • Wikipedia: In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole".[1] The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".

• Self-Similarity

  • Cynthia Lanius: Geometric figures are similar if they have the same shape. See animation.

  • Wikipedia: In mathematics, a self-similar object is exactly or approximately similar to a part of itself, e.g., the whole has the same shape as one or more of the parts.

• Fractal Dimension

  • Cynthia Lanius: A point has no dimensions ... A line has one dimension ...A plane has two dimensions ...Space, a huge empty box, has three dimensions ... Fractals can have fractional (or fractal) dimension

  • Wikipedia: In fractal geometry, the fractal dimension is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales.

  • List of fractals by Hausdorff dimension.

• Examples:

  • Willamette Math Emeritus Faculty - Junpei Sekino's Fractal page.
  • Fractal generator and Jackson Pollack painting
  • Examples
  • Fractal Mountains: Example 1 and Example 2.

• Generating Fractals

  • Complex Numbers:
    • Defined: Wikipedia, Dave's short course, many other web references are also available.
    • Applications
      • Question Corner: electric circuits, electromagnatism.
      • Wikipedia
  • Equations:
    • Mandelbrot set: For each point C in the complex plane, iterate: z(t+1) = z²(t) + C, where z(0) = 0
    • Julia set: Fix a value of C. For each point W in the complex plane, iterate: z(t+1) = z²(t) + C, where z(0) = W

• Povray and Fractals

  • See Julia Fractal in Povray online help.
  • Example Fractal Patterns
  • Fractal Examples generated using Povray.

  • These examples are largely taken from the online documentation in Povray:

    • Julia Set:
      and povray code: julia1.pov

    • Julia Set and Color Algorithms:
      and and and povray code: julia_e1i1.pov

    • Julia Set as Normals:
      and povray code: julia2.pov

    • Mandelbrot Set:
      and povray code: mandel1.pov

    • 3D slice of 4D Julia Set:
      and povray code: julia_fractal1.pov