Cynthia Lanius

Fractal Properties

fractal

Self-Similarity

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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

   
Cats, canaries, or kangaroos are similar if they are alike in some way. In geometry though, similar means something very specific. Geometric figures are similar if they have the same shape. I don't mean two rectangles or two triangles, but really the same shape. For example:

 
The two squares are similar.

squares
 
The two rectangles are not similar.

rectangles
 
But the two rectangles below are similar.

similar rectangles

Look carefully at the last blue rectangle and you will see that it is 2 times as wide as the red rectangle and 2 times as long. We say that the sides are in proportion and the ratio (or scale factor) is 2:1. Since the corresponding sides are in proportion (and the corresponding angles are also of equal measure), the figures are the same shape and are similar.

Consider similarity in another way. In order for one figure to be similar to another, you must be able to magnify the length of the small figure by the scale factor, and it will become exactly the same size as the larger figure.

Now how are figures self-similar?

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self-simMany figures that are not fractals are self-similar. Notice the figure to the right. Notice that the outline of the figure is a trapezoid. Now look inside at all the trapezoids that make up the larger trapezoid. This is an example of self similarity.

You can also think of self-similarity as copies. Each of the small trapezoids is a copy of the larger. Below are five other examples of self-similarity.

self-sim   self-sim      self-sim    self-sim   self-sim

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Self-Similarity of Fractals

Small SierpinskiTo the right is the Sierpinski Triangle that we make in this unit. Notice that the outline of the figure is an equilateral triangle. Now look inside at all the equilateral triangles. Remember that there are infinitely many smaller and smaller triangles inside. How many different sized triangles can you find? All of these are similar to each other and to the original triangle - self similarity

See all the copies of the original triangle inside? How many copies do you see where the ratio of the outer triangle's sides to the inner ones is 2:1? 4:1? 8:1? I think we have a pattern here. Can you find it?

Check out this very cool Sierpinski animated self-similarity illustration.

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Questions on Self-Similarity

Question 1: If the red image is the original figure, how many similar copies of it are contained in the blue figure?

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Question 2: Are squares self-similar? (Can you form bigger squares out of smaller ones?) Are hexagons? (Can you form larger hexagons out of smaller ones?) Draw examples to justify your answer.

 
Question 3:Are circles similar? Are they self-similar?(Can you form larger circles out of smaller ones? Draw examples to justify your answer.

 
Question 4: Experiment with designing another self-similar figure.

 

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Robert Devaney has more information on self-similarity.

You may obtain a print version of this page.

lanius@math.rice.edu

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