The title refers, of course, to Matthew (2:12) *''And being warned in a dream not to return to Herod, they departed to their own country by another  way''*. To be honest, it is not that specific particular case I'm more interested in. 

I'd like to have a reference, or a hint here, for a simple proof of the following fact (intuitive, but not easy to proof, as usual in these matters).

> Let $\Gamma$ be a simple Jordan arc in
> $\mathbb{R}^2$ (a homeomorphic image
> of the interval $[0,1]$). Then, $\Gamma$
> can be included in  a simple Jordan
> loop $ \Sigma $ (a homeomorphic image
> of  $\mathbb{S}^1$).

By the (generalized) Jordan's theorem, we know that $\mathbb{R}^2\setminus\Gamma$ is connected; and being open, it is even connected by piece-wise linear paths. The difficulty is that we need a path connecting the end-points of $\Gamma$; in other words, the question is how to show that $(\mathbb{R}^2\setminus\Gamma)\cup \partial\Gamma $ is path-connected (after that, an injective path could always be extracted). 

It seems to me everything would follow easily from this lemma:

> Assume that $B(0,2)\setminus \Gamma$
> has at least two connected components
> that meet $B(0,1)$. Then, there are
> three consecutive points of $\Gamma$,
> resp. $y_1$, $y_2$, and $y_3$  such
> that $\|y_2\|=1$, and 
> $\|y_1\|=\|y_3\|=2$

I'm also a bit puzzled by the quantitative aspect of this problem: 

> Assume that $\Gamma$ is parametrized
> by a homeomorphism
> $\gamma:[0,1]\to\Gamma$ with a modulus
> of continuity $\omega(t)$ (say, a
> continuous concave function vanishing
> at $t=0$)  and let $\omega_1$ be
> another modulus of continuity such
> that  $\omega_1(t) > \omega(t)$ for $t > 0$. 
> Is there a Jordan loop $\Sigma$ with parametrization $\gamma_1[0,2
> ]\to\Sigma$ that has modulus of
> continuity $\omega_1$ ?