7.13: Self-Similarity
Self-Similarity
When one part of an object can be enlarged (or shrunk) to look like the whole object it is self-similar.
To explore self-similarity, we will go through some examples. Typically, each step of a process is called an iteration. The first level is called Stage 0.
Sierpinksi 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.
Fractals
Like the Sierpinski triangle, a fractal is another self-similar object that is repeated at smaller scales. Below are the first three stages of the Koch snowflake.
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.
Examples
Example 1
Determine the number of edges and the perimeter of each snowflake shown in the Example titled "Fractals". Assume that the length of one side of the original (stage 0) equilateral triangle is 1.
Stage 0 | Stage 1 | Stage 2 | |
---|---|---|---|
Number of Edges | 3 | 12 | 48 |
Edge Length | 1 | \begin{align*}\frac{1}{3}\end{align*} | \begin{align*}\frac{1}{9}\end{align*} |
Perimeter | 3 | 4 |
\begin{align*}\frac{48}{9} = \frac{15}{3}\end{align*} |
Example 2
Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangle. Determine if there is a pattern.
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 \begin{align*}3: 3^0, 3^1, 3^2, 3^3, \ldots \end{align*}. The number of shaded triangles is the sum of the number of shaded and unshaded triangles from the previous stage. For Example, the number of shaded triangles in Stage 4 would equal 27 + 13 = 40.
Example 3
Determine the number of segments in each stage of the Cantor Set. Is there a pattern?
Starting from Stage 0, the number of segments is \begin{align*}1, 2, 4, 8, 16, \ldots\end{align*}. These are the powers of 2. \begin{align*}2^0, 2^1, 2^2,\ldots\end{align*}.
Review
- 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 | \begin{align*}\frac{1}{3}\end{align*} | \begin{align*}\frac{2}{3}\end{align*} |
Stage 2 | 4 | \begin{align*}\frac{1}{9}\end{align*} | \begin{align*}\frac{4}{9}\end{align*} |
Stage 3 | (2) | (3) | (4) |
Stage 4 | (5) | (6) | (7) |
Stage 5 | (8) | (9) | (10) |
- How many segments are in Stage \begin{align*}n\end{align*}?
- 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.
Review (Answers)
To view the Review answers, open this PDF file and look for section 7.13.
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Image Attributions
Here you'll learn what it means for an object to be self-similar and you'll be introduced to some common examples of self-similarity.
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