Cynthia Lanius

Fractal Properties

Fractal Dimension

 
Table of Contents

Introduction

  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
         Anti-Snowflake
            Using Java

  Fractal Properties
    Self-similarity
    Fractional dimension
    Formation by iteration

  For Teachers
    Teachers' Notes
    Teacher-to-Teacher

  Comments
    My fractals mail
    Send fractals mail

  Fractals on the Web
    The Math Forum

  Other Math Lessons
    by Cynthia Lanius

  Awards
    This Site has received

   
A point has no dimensions - no length, no width, no height.
That dot is obviously way too big to really represent a point. But we'll live with it, if we all just agree what a point really is.
 
A line has one dimension - length. It has no width and no height, but infinite length.

Again, this model of a line is really not very good, but until we learn how to draw a line with 0 width and infinite length, it'll have to do.
 
A plane has two dimensions - length and width, no depth.

It's an absolutely flat tabletop extending out both ways to infinity.
 
Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions.
 
Obviously this isn't a good representation of 3-D. Besides its size, it's just a hexagon drawn to fool you into thinking it's a box.
  
Fractals can have fractional (or fractal) dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate below.

 

Just as the images above weren't very good pictures of a point, line, plane, or space, the drawing meant to be the Sierpinski Triangle has limitations. Remember as we continue that fractals are really formed by infinitely many steps. So there are infinitely many smaller and smaller triangles inside the figure, and infinitely many holes (the black triangles).

Let's look further at what we mean by dimension. Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment.

 

Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies.

Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do you get? Doubling the side gives eight copies.
Let's organize our information into a table.

 Figure  Dimension  No. of Copies 
Line segment
1
2 = 21
Square
2
4 = 22
Cube
3
8 = 23

Do you see a pattern? It appears that the dimension is the exponent - and it is! So when we double the sides and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension.

Let's add that as a row to the table.

 Figure  Dimension  No. of Copies 
Line Segment
1
2 = 21
Square
2
4 = 22
Cube
3
8 = 23
 Doubling Similarity 
d
n = 2d

We can use this to figure out the dimension of the Sierpinski Triangle because when you double the length of the sides, you get another Sierpinski Triangle similar to the first.
  

Start with a Sierpinski triangle of 1-inch sides. Double the length of the sides. Now how many copies of the original triangle do you have? Remember that the black triangles are holes, so we can't count them.

Doubling the sides gives us three copies, so 3 = 2d, where d = the dimension.

But wait, 2 = 21, and 4 = 22, so what number could this be? It has to be somewhere between 1 and 2, right? Let's add this to our table.

 Figure   Dimension  No. of Copies 
Line Segment
1
2 = 21
Sierpinski's Triangle
?
3 = 2?
Square
2
4 = 22
Cube
3
8 = 23
 Doubling Similarity 
d
n = 2d

So the dimension of Sierpinski's Triangle is between 1 and 2. Do you think you could find a better answer? Use a calculator with an exponent key (the key usually looks like this ^ ). Use 2 as a base and experiment with different exponents between 1 and 2 to see how close you can come. For example, try 1.1. Type 2^1.1 and you get 2.143547. I'll bet you can get closer to 3 than that. Try 2^1.2 and you get 2.2974. That's closer to 3, but you can do better.

Okay, I got you started; now find the exponent that gets you closest to 3, and that's its dimension. Email me your answer if you want.

lanius@rice.edu

That's how fractals can have fractional dimension.

Note 1: For those of you that know about logarithms: Yes, you could use logs to solve this, but remember, this is primarily designed for students in elementary and middle school who haven't studied them yet. If you have studied logarithms, then yes, use them to solve this equation.

Note 2: Is there anything special about us doubling the lengths here? Could you have tripled them and derived the formula as well? Why don't you investigate by trying it.

Note 3: Robert Devaney has more information on fractal dimension.

Note 4: You may obtain a print version of this page.

Copyright 1997-2007 Cynthia Lanius
URL http://math.rice.edu/~lanius/fractals/dim.html