FRACTALS UNLEASHED
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Tutorial Chapter 8

Intro to Iteration

WHAT IS IT?

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:

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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.

There are also several minor types of iteration. For example, fractals can be made by iterating the process of folding paper. However, fractals like that can also be created using at least one of the main types of iteration.

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