Sequences - fractals and geometric sequences

The following links provide pictures of the different stages of a fractal. Your task is to develop a formula for the general term t_n where t_n is the value for the n-th picture. Assume that n=1 for the starting picture.

Note - The n-th picture is the result of n-1 steps or iterations.

Cantor's Comb

• If the fraction removed is 1/3, what is the total length of the lines in the n-th picture?
• If the fraction removed is f, what is the total length of the lines in the n-th picture?

• What is the total length of the lines in the n-th picture?

• What is the total length of the lines in the n-th picture?

• What is the area of the n-th picture?

• What is the total of the perimeters of the shaded triangles in the n-th picture?
• What is the total area of the shaded triangles in the n-th picture?

• Using the grid provided, draw the 2nd, 3rd and 4th pictures.
• What is the perimeter of the n-th picture?
• What is the area of the n-th picture?
• Explain why the Koch Snowflake has an infinite perimeter and a finite area.

Other links:

• Sierpinski Triangle activities by Cynthia Lanius
• Sierpinski Gasket by Paul Bourke (includes 3 dimensions)
• Waclaw Sierpinski Biography
• Koch Snowflake activities by Cynthia Lanius (includes Anti-Snowflake)

Design your own fractal carpet and find a formula for the area of the n-th picture? (The two examples were drawn with Coreldraw.)

• Example 1
• Example 2 (Sierpinski Triangle)

The image opposite was produced with the Pascal's Triangle Interface. It shows rows of Pascal's triangle with odd numbers coloured red. Which Sierpinski Triangle picture matches this image? Use the Pascal's Triangle Interface to produce the image matching the next Sierpinski Triangle picture.