Suppose that we draw each of these fractals at level ℓ and with size 1. What is the perimeter of each of these fractals? (By “perimeter,” we mean the total length of all boundaries separating regions...

Suppose that we draw each of these fractals at level ℓ and with size 1. What is the perimeter of each of these fractals? (By “perimeter,” we mean the total length of all boundaries separating regions inside the figure from regions outside— which includes, for example, the boundary of the “hole” in the Sierpinski carpet. For the Sierpinski fractals as drawn here, the perimeter is precisely the length of lines separating colored-in from uncolored-in regions.) In each case, conjecture a formula and prove your answer correct by induction.

1.Von Koch snowflake

2. Sierpinski triangle

3. Sierpinski carpet