I've been considering for a while the problem of non rectifiable curves. For example, my book showed me a curve defined such as If $$ f:[0, 1] \to \mathbb R$$ $$f(t)= t\sin\left(\frac {\pi} {2t}\right), f(0)=0$$ and $$ \gamma: [0,1] \to \mathbb R^2, \gamma(t)=(t, f(t)) $$ and led me to prove it is has an infinite length, after a long argument (with polygonal chains and inequalities).
Then I started looking for other examples, and came across the famous Koch's and Peano's curves, but didn't find many of them.
I'm wondering:
Do you know other curves which are not rectifiable, preferably with an explicit expression, when it exists (such as the one I wrote beforehand)? Is there a "list" of the most important ones?
The argument showing that your $\gamma$ has infinite length need not be long. For any $N\in \mathbb N,$ the length will be at least
$$\sum_{n=1}^{N}|\gamma(1/2n)-\gamma(1/(2n-1))| \ge\sum_{n=1}^{N}|f(1/2n)-f(1/(2n-1))|$$ $$ = \sum_{n=1}^{N}|1/(2n)\cdot\sin(n\pi) - 1/(2n-1)\cdot \sin((2n-1)\pi/2) | = \sum_{n=1}^{N}=1/(2n-1).$$
The last sum $\to \infty$ as $N\to \infty.$ Thus the length of $\gamma$ is $\infty.$