How many times line segments can intersect a Jordan curve? I posted a question on math.stackexchange.com but it seems this question might be open
https://math.stackexchange.com/questions/109752/line-segments-intersecting-jordan-curve
Namely, 
is there a set $A\subset \mathbb{R}^2$ such that 


*

*The boundary of $A$, $\partial A$, is a Jordan curve and

*For any $B\in \operatorname{int} A\ne\emptyset $, $C\in \operatorname{ext} A\ne\emptyset$ , the line segment $BC$ intersects $\partial A$ infinitely many times?


In the link Leonid Kovalev gave an example that might solve the problem but I have no idea is that an example of such curve. Can anyone verify if the Julia set he gave in the link solves the problem?
 A: Most* of Jordan curves have this property. 
Moreover for most of curves you can not see a point on curve from one side if you can see it from the other side. 
To construct such example, start with a smooth closed curve $\gamma_0$, 
note that you can wave it,
so it will remain smooth and any point which is visible from one side on distance 1 
is not visible from the other side on distance 1.

Given $\varepsilon_1>0$, you may assume in addition that the new curve $\gamma_1$ is $\varepsilon_1$-close to $\gamma_0$; i.e. 
$$|\gamma_1(t)-\gamma_0(t)|<\varepsilon_1$$
for any $t$.
Repeat this procedure for distances $\tfrac12$, $\tfrac13$ and so on.
At each step, choose $\varepsilon_n$ very small, depending on  $\gamma_{n-1}$,
then the limit $\gamma_\infty$ is a Jordan curve you want.
$*$ "Most" means a G-delta dense set of Jordan curves. 
A: I might be wrong, but the Newton fractal:
http://en.wikipedia.org/wiki/File:Julia_set_for_the_rational_function.png
might be a good candidate.
See also http://en.wikipedia.org/wiki/Lakes_of_Wada
