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 self-similarity to answer questions like this one.
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CK-12 Foundation: Self-Similarity
Guidance
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.
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 self-similar 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.
CK-12 Foundation: Self-Similarity
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 | \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*} |
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 \begin{align*}3: 3^0, 3^1, 3^2, 3^3, \ldots \end{align*}. The number of shaded triangles is the sum the 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.
3. 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*}.
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 | \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 | |||
Stage 4 | |||
Stage 5 |
- How many segments are in Stage \begin{align*}n\end{align*}?
- 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.