7.7: Extension : SelfSimilarity
Learning Objectives
 Understand basic fractals.
SelfSimilar: When one part of an object can be enlarged (or shrunk) to look like the whole object.
To explore selfsimilarity, we will go through some examples. Typically, each step of repetition is called an iteration. The first level is called Stage 0.
Sierpinski Triangle
The Sierpinski triangle iterates a triangle by connecting the midpoints of the sides and shading the central triangle (Stage 1). Repeat this process for the unshaded triangles in Stage 1 to get Stage 2.
Example 1: Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangles. Determine if there is a pattern.
Solution:
Stage 0  Stage 1  Stage 2  Stage 3  

Unshaded  1  3  9  27 
Shaded  0  1  4  13 
The unshaded triangles seem to be powers of
Fractals
A fractal is another selfsimilar object that is repeated at smaller scales. Below are the first three stages of the Koch snowflake.
Example 2: Determine the number of edges and the perimeter of each snowflake.
Stage 0  Stage 1  Stage 2  

Number of Edges  3  12  48 
Edge Length  1 


Perimeter  3  4 

The Cantor Set
The Cantor set is another fractal that consists of dividing a segment into thirds and then erasing the middle third.
Review Questions
 Draw Stage 4 of the Cantor set.
 Use the Cantor Set to fill in the table below.
Number of Segments  Length of each Segment  Total Length of the Segments  

Stage 0  1  1  1 
Stage 1  2 


Stage 2  4 


Stage 3  
Stage 4  
Stage 5 
 How many segments are in Stage
n ?  Draw Stage 3 of the Koch snowflake.
 A variation on the Sierpinski triangle is the Sierpinski carpet, which splits a square into 9 equal squares, coloring the middle one only. Then, split the uncolored squares to get the next stage. Draw the first 3 stages of this fractal.
 How many colored vs. uncolored square are in each stage?
 Fractals are very common in nature. For example, a fern leaf is a fractal. As the leaves get closer to the end, they get smaller and smaller. Find three other examples of fractals in nature.
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