7.13. Self-Similarity http://www.ck12.org
Practice
- 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
or another field that interests you.
Summary
This chapter is all about proportional relationships. It begins by introducing the concept of ratio and proportion and
detailing properties of proportions. It then focuses on the geometric relationships of similar polygons. Applications
of similar polygons and scale factors are covered. The AA, SSS, and SAS methods of determining similar triangles
are presented and the Triangle Proportionality Theorem is explored. The chapter wraps up with the proportional
relationships formed when parallel lines are cut by a transversal, similarity and dilated figures, and self-similarity.
Chapter Keywords
- Ratio
- Proportion
- Means
- Extremes
- Cross-Multiplication Theorem
- Similar Polygons
- Scale Factor
- AA Similarity Postulate
- Indirect Measurement
- SSS Similarity Theorem
- SAS Similarity Theorem
- Triangle Proportionality Theorem
- Triangle Proportionality Theorem Converse