An

iterated

function system (

IFS) is a function or

set of functions which operate on

data repeatedly. Often, the

input and/or functions are kept

simple so as to be more easily analyzed. Some

examples of IFS include the

dragon fractal,

Sierpinski triangle,

Koch curve,

L-systems and the

Mandelbrot set. By way of example, below is the Sierpinski

triangle followed by another similar

fractal shown step by

step (for three steps each):

________________
| |
| |_ |
| |
| |
| |
| |
| |
| |
----------------
________________________________
| || |
| |_ || |_ |
| || |
| || |
| || |
| || |
| || |
| || |
--------------------------------
________
| |
| |
| |
| |
--------
________________
| || |
| || |
| || |
| || |
----------------
________ ________
| | | |
| | | |
| | | |
| | | |
-------- --------
________________________________
| || || || |
| || || || |
| || || || |
| || || || |
--------------------------------
____
| |
| |
----
________
| || |
| || |
--------
____ ____
| | | |
| | | |
---- ----
________________
| || || || |
| || || || |
----------------
____ ____
| | | |
| | | |
---- ----
________ ________
| || | | || |
| || | | || |
-------- --------
____ ____ ____ ____
| | | | | | | |
| | | | | | | |
---- ---- ---- ----
________________________________
| || || || || || || || |
| || || || || || || || |
--------------------------------

________________
| _ |
| | |
| |
| |
| |
| |
| |
| |
----------------
________________________________
| || |
| |_ || |
| || |
| || |
| || |
| || |
| || _| |
| || |
--------------------------------
________________
| || |
| || |
| || |
| || |
----------------
________
| |
| |
| |
| |
--------
________ ________
| | | |
| | | |
| | | |
| | | |
-------- --------
________________________________
| || || || |
| || || || |
| || || || |
| || || || |
--------------------------------
________________
| || || || |
| || || || |
----------------
____ ____
| | | |
| | | |
---- ----
____
| |
| |
----
________
| || |
| || |
--------
________ ________
| || | | || |
| || | | || |
-------- --------
____ ____
| | | |
| | | |
---- ----
____ ________ ____
| | | || | | |
| | | || | | |
---- -------- ----
________________________________
| || || || || || || || |
| || || || || || || || |
--------------------------------

With the three-way IFS above, many of the fractals that are produced are simple

reflections or restatements of each other. Out of 512 possible fractals, only 200 or less are

unique for this particular IFS form.

*I worked in the extra spaces above to make sure that the overall idea is clear.*