After staring at a small portion of a Koch snowflake, it seems to me that it is not possible to tell the inside of the snowflake from the outside if you can only see a small portion of the boundary.
Small view of the Koch snowflake:
Expanded view:
Problem: Given a closed fractal curve and a neighborhood of a point on this curve, we would like to identify which side of the curve is the interior.
Question: Are there useful conditions on the fractal curve which tell us when this problem is solvable?
Edit: I realized there is one way to tell the interior of the Koch snowflake from the outside, with additional assumptions: if the interior is formed from triangles of side lengths $1, 1/3, 1/9, \dots$, then the exterior has triangular features of size $1/\sqrt{3}, 1/3\sqrt{3}, 1/9\sqrt{3}, \dots$. So if you have a ruler (i.e. metric) and you know the overall size of the snowflake, you can determine locally which side is the interior.