Tutorial Chapter 8   

Intro to Iteration

Any fractal has some infinitely repeating pattern. When creating such fractal, you would suspect that the easiest way is to repeat a certain series of steps which create that pattern. Instead of the word "repeat" we use a mathematical synonym "iterate" and the process is called iteration. In fact, any fractal can be made by the iteration of a certain rule. For example, the rule for creating the Koch Snowflake is:

To create a true fractal, we have to iterate an infinite amount of times. However, when doing it on a computer, we are limited by speed and resolution, so we only iterate a certain number of times. Increasing the number of iterations makes the fractal more accurate.

TYPES OF ITERATION

There are three basic types of iteration:

1. Generator Iteration — Creates fractals by repeatedly substituting certain geometric shapes with other shapes.
2. IFS Iteration — Creates fractals by repeatedly applying geometric transformations (such as rotation and reflection) to points.
3. Formula Iteration — Includes several ways of creating fractals by repeating a certain mathematical formula or several formulas.