7.7: Extension: Self-Similarity
Learning Objectives
- Draw sets of the Sierpinski Triangle.
- Understand basic fractals.
Self-Similar: When one part of an object can be enlarged (or shrunk) to look like the whole object.
To explore self-similarity, we will go through a couple of examples. Typically, each step of repetition is called an iteration or level. The first level is called the Start Level or Stage 0.
Sierpinski Triangle
The Sierpinski triangle iterates an equilateral triangle (but, any triangle can be used) 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. This series was part of the Know What? in Section 5.1.
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 \begin{align*}3, 3^0, 3^1, 3^2, 3^3, \ldots\end{align*} The shaded triangles are add the previous number of unshaded triangles to the total. For Example, Stage 4 would equal \begin{align*}9 + 13\end{align*} shaded trangles.
Fractals
A fractal is another self-similar object that is repeated at successively 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 | \begin{align*}\frac{1}{3}\end{align*} | \begin{align*}\frac{1}{9}\end{align*} |
Perimeter | 3 | 4 | \begin{align*}\frac{48}{9}=5.\overline{3}\end{align*} |
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 | \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*}?
- What is the length of each segment in Stage \begin{align*}n\end{align*}?
- Draw Stage 3 of the Koch snowflake.
- Fill in the table from Example 2 for 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 shaded vs. unshaded squares 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.
- 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.