Fractal Properties
Why study fractals? Making fractals
Fractal Properties
For Teachers
Comments
Fractals on the Web
Other Math Lessons
Awards

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:
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 selfsimilar? 
Many figures that are not fractals are selfsimilar. 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 selfsimilarity as copies. Each of the small trapezoids is a copy of the larger. Below are five other examples of selfsimilarity.
SelfSimilarity of FractalsURL http://math.rice.edu/~lanius/fractals/self.htmlTo 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 selfsimilarity illustration.
Questions on SelfSimilarity
Question 2: Are squares selfsimilar? (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 1: If the red image is the original figure, how many similar copies of it are contained in the blue figure?
Question 3:Are circles similar? Are they selfsimilar?(Can you form larger circles out of smaller ones? Draw examples to justify your answer.
Question 4: Experiment with designing another selfsimilar figure.
Robert Devaney has more information on selfsimilarity. You may obtain a print version of this page.
Copyright 19972007 Cynthia Lanius