Tutorial Chapter 12
AND YET ANOTHER TYPE OF ITERATION
Formula iteration is the simplest type of iteration, yet it is the most important and creates the most complex patterns. It is based on taking a number and repeatedly substituting it with another one using a mathematical formula. Three types of fractals can be created by using different kinds of formula iteration:
1. Strange Attractors — simply start with coordinates of an initial
point and determine every next point using the formula.
2. Julia Sets — start with coordinates of some point. Color it depending on what happens to it after iteration. Do the same for every point.
3. Mandelbrot Set — start with the point (0, 0). Use the formula with some other point’s coordinates used as constants in the formula. Color that point depending on what happens to (0, 0) after iteration. Do the same for every point.
THE COOL STUFF
Generally, iterating a formula changes a number. This is used to study things around us that exhibit change, such as chemical reactions, population, and the weather. Very simple formulas are capable of creating some of the most complex fractals. In the next lesson, you will be introduced to why this happens.
Cosmos : The Art of Mathematical Design
You will not find a lot of explanations in this book, but it is great if you want to see the great variety and beauty of fractals, especially formula generated ones. Every fractal in the book is followed by a formula and parameters, letting you experiment on your own.