Boundary length

An important property of the Koch Snowflake is that its boundary has infinite length. This is especially suprising in light of the fact that the snowflake encloses only a finite area (after all, it can be completely covered with a square of paper).

To show that the boundary of the snowflake has infinite length, it suffices to show that each of the three congruent fractals making up the snowflake has infinite length. Suppose that the initial segment (call it K0) has length 1. Then K1, the curve produced by removing the middle one-third of K0 and adding two new segments having the same length, has length 4/3. The curve K2 at the end of the second stage has length 42/32. Repeating this process, the curve Kn produced after n stages has length 4n/3n. Hence, the length of the limit curve K is given by the following cell.

Input := 

Limit[(4/3)^n,n->Infinity]

Output =

Infinity