Cynthia Lanius

Fractal Properties

Iterative Formation

Table of Contents


  Why study fractals?
    What's so hot about
    fractals, anyway?

  Making fractals
    Sierpinski Triangle
         Using Java
         Math questions
         Sierpinski Meets Pascal
    Jurassic Park Fractal
         Using JAVA
         It grows complex
         Real first iteration
         Encoding the fractal
         World's Largest
    Koch Snowflake
         Using Java
         Infinite perimeter
         Finite area
            Using Java

  Fractal Properties
    Fractional dimension
    Formation by iteration

  For Teachers
    Teachers' Notes

    My fractals mail
    Send fractals mail

  Fractals on the Web
    The Math Forum

  Other Math Lessons
    by Cynthia Lanius

    This Site has received

Fractals are often formed by what is called an iterative process. Here's what I mean.

To make a fractal: Take a familiar geometric figure (a triangle or line segment, for example) and operate on it so that the new figure is more "complicated" in a special way.

Then in the same way, operate on that resulting figure, and get an even more complicated figure.

Now operate on that resulting figure in the same way and get an even more complicated figure.

Do it again and again...and again. In fact, you have to think of doing it infinitely many times.

You can observe this iterative process in all the fractals that we make in this unit:

Not every iterative process produces a fractal. Take a line segment and chop off the end. What is the resulting figure? Just another line segment - not "complicated" at all, and not a fractal. You could continue the iterative process over and over, chopping off the end of the line segment, but it would just become a shorter and shorter line segment, not "complicated", not fractal.

Below is a picture of a similar iterative operation that is fractal. Take a line segment (see below) and remove the middle third. What is the resulting figure? Hmmm. That's a more complicated figure. It's a line segment with a hole in it.

Repeat the process on that figure. In other words, remove the middle third of both of those sections. This produces an even more complicated figure. Now think of doing this infinitely many times. In fact, this is a famous fractal called Cantor Dust.

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Copyright 1997-2007 Cynthia Lanius