Lecture/Lab 11: Fractals

R. Robinson 11/99, 4/01
 

1. Here is a website devoted to fractals in Maple.

At the bottom of the first page is a Maple workspace you can download that has these examples in it.
You can try some Julia and Mandelbrodt sets ans some the IFS images here (as well as some other things--I couldn't find any L-systems).

2. One of the best fractals programs ever written is Fractint. There are various versions. For a PC the DOS version is reportedly better than the Windows version (form a mathematical point of view). This program can draw all the kinds of fractals we have discussed in class. http://spanky.triumf.ca/www/fractint/fractint.html

3. Here are two web sites that implement L-systems as java applets.

I was not able to find any Maple or MATLAB implementations of L-systems, (but writing one might make a good project)!

A good reference for this subject is Przemyslaw Prusinkiewicz, Aristid Lindenmayer, The Algorithmic Beauty of Plants (The Virtual Laboratory) Springer Verlag, 1990.

4. Here are some Matlab M-files that implement iterated function systems ifs.m.

This is based on a useful trick for expressing an affine transformation f(x)=Ax+b on R2 as multiplication by a 3X3 matrix. In particular if x=(x,y)t put x'=(x,y,1)t. Put T=[A; 0 0 1] (MATLAB notation). Then if y'=Tx', y=(f1(x),f(x),1), i.e. you get the affine tranformation by chopping off the third coordinate.

Try the following matrices to get the Barnsley fern:

T1=[.86 .86 0; -.03 .86 1.5; 0 0 1];
T2=[.2 -.25 0; .21 .23 1.5; 0 0 1];
T3=[-.15 .27 0; .25 .26 .45; 0 0 1];
T4=[0 0 0; 0 0 .17; 0 0 1];

Can you get the Sirpinski carpet?

A good reference on iterated function systema is Michael Barnsley, Fractals Everywhere, Morgan Kaufmann, 1993.