Guidance
When one part of an object can be enlarged (or shrunk) to look like the whole object it isself-similar.
To explore self-similarity, we will go through some examples. Typically, each step of a process is called aniteration.The first level is calledStage 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.
Vocabulary
When one part of an object can be enlarged (or shrunk) to look like the whole object it isself-similar.
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 2Number of Edges / 3 / 12 / 48
Edge Length / 1 / /
Perimeter / 3 / 4 /
2.
Stage 0 / Stage 1 / Stage 2 / Stage 3Unshaded / 1 / 3 / 9 / 27
Shaded / 0 / 1 / 4 / 13
The number of unshaded triangles seems to be powers of. 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. These are the powers of 2..