# 7.7: Extension : Self-Similarity

Difficulty Level: At Grade Created by: CK-12

## 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

1. Draw Stage 4 of the Cantor set.
2. 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
1. How many segments are in Stage \begin{align*}n\end{align*}?
2. Draw Stage 3 of the Koch snowflake.
3. 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.
4. How many colored vs. uncolored square are in each stage?
5. 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.

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