posted by alex .
you start with an equilateral triangle and then trisect each of the sides with a new equilateral triangle on the middle third of each trisection, repeat the process indefinitely
how would you prove that the perimeter is infinite?
You start out with three sides of length a, and the figure perimeter is 3a.
After the first step you described, you have a figure with 12 sides of length a/3. The perimeter is now 4a.
After a similar second step, you will have 48 sides of length a/9. The perimeter is now 5.33 a
After N such steps, you have 3*4^N sides of length a/3^N, with a perimeter of = 3a*(4/3)^N
That number clearly becomes infinite as N becomes infinite