Chapter 11.6, Problem 12P

To determine

To prove: the Koch curve that described has no perimeter or unbounded length.

Explanation of Solution

Given:

  

Concept used:

According to the Limits formula:

  $\begin{array}{l}\underset{n\to \infty }{\lim }{\left(x\right)}^n=\infty .\\ \underset{n\to \infty }{\lim }{\left(x\right)}^{-n}=0.\end{array}$

Calculation:

In each stage a third of the perimeter is removed and then $\frac{3}{3}$ of the perimeter is added by adding $3$ third each side.

The perimeter is thus increased by a factor $\frac{5}{3}$ .

Perimeter $=\underset{n\to \infty }{\lim }4{\left(\frac{5}{3}\right)}^n=\infty$ .

Hence, the perimeter is $\infty$ .