Fractals Everywhere

So "God Does Play Dice"

Introduction

Welcome to my fractal page. Because there are already so many fractal pages out there, I will only provide on this page the cream of the available fractal resources on the 'net. I hope you find this page a nice introduction to fractals, and come to fall in love with these beautiful mathematical objects as I did one day. So explore, and don't be afraid to wonder off to the other great fractal pages!

There are many ways in which fractals can be generated. All fractals are generated by repeatedly carrying out a series of calculations, called iterations. The L-System Fractals, the easiest to create by hand, are made by repeatedly replacing a shape or character by another shape or character. Look at the L-System Tree, and note how each branch is replaced by two others at certain distance apart. Now, each new branch is replaced like the previous one, and so on. IFS fractals are more complicated to create. Although the Sierpinski Triangle may be look like it is being generated using the L-System approach, a closer examination reveals that it is in fact generated by "randomly" transforming points based on a transformation mechanism called a frame. Each point on the screen is placed in a predefined "random" frame, and the image is crafted in this way. The most beautiful and complex fractals are generated in a more complicated fashion. Each point on the screen is run through a formula, and the result after a series of hundreds of iterations is examined. If the result stays within the screen dimensions, it is colored black. If the points bounds to infinity (by going off the screen), that point is colored by the color determined by the amount of iterations it took for the point to go off the screen.

Note how in the first image the point currently being calculated does not leave the screen, indicating that the point must be "within" the set, whereas in the second image the point goes off to infinity.




Fractal Art

Sierpinski Triangle Sierpinski Square
Koch Snowflake L-System Tree
L-System Tree (wide) L-System Tree (thin)
Complex Fractals



Fractal Java Simulator




Fractal Noise

Brown

White

Pink


Further Reading

Yahoo Fractal Page

Fantastic Fractals

Exploring Chaos: A Guide to the New Science of Disorder by Nina Hall

Fractals Everywhere by Michael Barnsley

Fractals: The Patterns of Chaos by John Briggs


Copyright © 1998, by Dan Hussain
All rights reserved, unless otherwise stated.