Existence of an Infinite Length Path I came across the following simple definition

A path $\gamma$ in $\mathbb{R}^n$ that connects the point $a \in \mathbb{R}^n$ to the point $b \in \mathbb{R}^n$, is a continuous $\gamma : [0, 1] \to \mathbb{R}^n$ such that $\gamma(0) = a$ and $\gamma(1) = b$. We denote by $\ell(\gamma)$ the (Euclidean) length of $\gamma$. $\ell(\gamma)$ is always defined and is either a non-negative realnumber or $\infty$.

However, I cannot seem to think of a path, defined in this manner (specifically, where the domain is compact), the length of which is infinite.
Can anyone provide an example ?

Yes, for instance Koch snowflake is a such example. Let's do the following construction:


*

*Start with the segment $A_0 = [0,1]$.

*Subdivise $A_0$ in three equal pieces.

*Replace the middle third by an equilateral triangle with base $[\frac13, \frac23]$.

*Suppress the base of that triangle. You get the path $A_1$ something which looks like a saw teeth.

*By induction, to construct $A_{n+1}$ replace each of the $4^n$ segments of $A_n$ by an equilateral triangle of base this segment, and then remove the segment.

*The limit object is a path (whose image is compact set) joining $0$ and $1$.


The length of $A_n$ is $\left( \frac43 \right)^n$ tends to $+\infty$ as $n \to +\infty$.
The Hausdorff dimensions is $\frac{\ln 4}{\ln 3} \approx 1.26$
The following picture summarize the construction.
http://www.cl.cam.ac.uk/~dao29/tmp/koch/geometric-construction.png
