7.7: Extension : Self-Similarity
Learning Objectives
- 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 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 \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 9 + 13 shaded triangles.
Fractals
A fractal is another self-similar 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 | \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*}?
- 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.