CK-12 Geometry Concepts

http://www.ck12.org Chapter 7. Similarity

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.

Watch this video for help with the Examples above.

MEDIA

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CK-12 Foundation: Chapter7SelfSimilarityB

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:

TABLE7.3:

Stage 0 Stage 1 Stage 2

Number of Edges 3 12 48

Edge Length (^11319)
Perimeter 3 4 489 =^153

2.

TABLE7.4:

Stage 0 Stage 1 Stage 2 Stage 3

Unshaded 1 3 9 27

Shaded 0 1 4 13

The number of unshaded triangles seems to be powers of 3 : 3^0 , 31 , 32 , 33 ,.... 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 1, 2 , 4 , 8 , 16 ,.... These are the powers of 2. 2^0 , 21 , 22 ,....