Fractals

Tom Kelliher, CS 320

Mar. 11, 1998

Announcements: pong due Friday.

From last time:

1. Pong

Outline:

1. Preliminaries.

2. Fractals.

3. Julia-Fatou sets.

4. Mandelbrot set.

Assignment: Read Section 9.5.

Preliminaries

1. : the ``imaginary'' number.

2. Complex numbers.

3. The complex plane.

   Representation of in the complex plane:

   

4. Complex addition:

   

5. Complex multiplication:

   

6. Complex magnitude:

   

   (Cartesian distance)

Fractals

1. Fractal: Any object which has a substantial measure of exact or statistical self-similarity. Precisely speaking, true fractals have statistical self-similarity at all resolutions.

2. Self-similar object: An object which can be translated, rotated, and scaled onto a sub-portion of itself.

3. ``Self-similar'' objects: coasts, mountains, trees, plants.

4. von Koch snowflake example:

   

   Scale snowflake by 1/3 and replace each of four ``segments'' with scaled snowflake.

Julia-Fatou Sets

1. Consider the complex sequence:

   

   where is a point in the complex plane and c is a complex constant.

2. What is the behavior if c=0?

3. What can happen with the sequence?
   1. Be ``attracted'' to a finite value.

   2. Be attracted to infinity.

   3. Not be attracted to any value (diverge).

   The Julia-Fatou set is this latter set of values.

4. Structure of Julia-Fatou sets: some simple, some complex.

5. The notion of ``connectivity.''

Mandelbrot Set

1. The Julia-Fatou set associated with a complex number.

2. Definition: The Mandelbrot set consists of those complex numbers whose associated Julia-Fatou set is ``connected.''

3. Consider computing the Mandelbrot set.

4. An approximation. Consider the sequence:

   

   where z is a point in the complex plane which we're considering for inclusion in the Mandelbrot set. Notes:

   1. Usually run 100--1,000 iterations.

   2. The point's color is chosen as a function of how many iterations are completed before , where t is a threshold value.

   3. Increasing the number of iterations and/or the threshold improves the approximation.

   4. Typical values: 100 iterations, threshold of 2.