7.13. Self-Similarity http://www.ck12.org
- Determine the number of shaded and unshaded triangles in each stage of the Sierpinkski triangle. Determine if
there is a pattern. - 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 48Edge 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 13The number of unshaded triangles seems to be powers of 3 : 3^0 , 31 , 32 , 33 ,.... The number of shaded triangles is the
sum of 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.
- Starting from Stage 0, the number of segments is 1, 2 , 4 , 8 , 16 ,.... These are the powers of 2. 2^0 , 21 , 22 ,....
Explore More
- Draw Stage 4 of the Cantor set.
Use the Cantor Set to fill in the table below.
TABLE7.5:
Number of Segments Length of each Segment Total Length of the Seg-
ments
Stage 0 1 1 1Stage 1 (^21323)
Stage 2 (^41949)
Stage 3 (2) (3) (4)
Stage 4 (5) (6) (7)
Stage 5 (8) (9) (10)
- How many segments are in Stagen?
- What is the total length of the segments in Stage n?.
- 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 squares are in each stage?
- Use the internet to explore fractals further. Write a paragraph about another example of a fractal in music, art