<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation

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.


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.


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.

Image Attributions

Show Hide Details
Files can only be attached to the latest version of section
Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original

Original text