7.13: SelfSimilarity
What if you were given an object, like a triangle or a snowflake, in which a part of it could be enlarged (or shrunk) to look like the whole object? What would each successive iteration of that object look like? After completing this Concept, you'll be able to use the idea of selfsimilarity to answer questions like this one.
Watch This
CK12 Foundation: Chapter7SelfSimilarityA
Brain Waves: Sierpinski Triangle
Guidance
When one part of an object can be enlarged (or shrunk) to look like the whole object it is selfsimilar.
To explore selfsimilarity, we will go through some examples. Typically, each step of a process is called an iteration. The first level is called Stage 0.
Example A (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 B (Fractals)
Like the Sierpinski triangle, a fractal is another selfsimilar object that is repeated at smaller scales. Below are the first three stages of the Koch snowflake.
Example C (The Cantor Set)
The Cantor set is another example of a fractal. It consists of dividing a segment into thirds and then erasing the middle third.
Watch this video for help with the Examples above.
CK12 Foundation: Chapter7SelfSimilarityB
Vocabulary
When one part of an object can be enlarged (or shrunk) to look like the whole object it is selfsimilar.
Guided Practice
1. Determine the number of edges and the perimeter of each snowflake shown in Example B. Assume that the length of one side of the original (stage 0) equilateral triangle is 1.
2. Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangle. Determine if there is a pattern.
3. Determine the number of segments in each stage of the Cantor Set. Is there a pattern?
Answers:
1.
Stage 0  Stage 1  Stage 2  

Number of Edges  3  12  48 
Edge Length  1 


Perimeter  3  4 

2.
Stage 0  Stage 1  Stage 2  Stage 3  

Unshaded  1  3  9  27 
Shaded  0  1  4  13 
The number of unshaded triangles seems to be powers of
3. Starting from Stage 0, the number of segments is
Practice
 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  (2)  (3)  (4) 
Stage 4  (5)  (6)  (7) 
Stage 5  (8)  (9)  (10) 
 How many segments are in Stage
n ?  What is the total length of the segments in Stage n?.
 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 squares are in each stage?
 Use the internet to explore fractals further. Write a paragraph about another example of a fractal in music, art or another field that interests you.
Image Attributions
Here you'll learn what it means for an object to be selfsimilar and you'll be introduced to some common examples of selfsimilarity.