Start with a simple non-self intersecting curve on the plane from point $A$ to $B$. Then add a translated copy of that curve next to it, so they from a new curve, and assume that also the new curve does not self-intersect.
My question is: if I iterate this indefinitely, can intersections appear (given the first and second copy did not intersect)?
This is certainly not true in 3D space. I can make a curve and concatenate three copies, so that the first and second (and the second and third) copy do not intersect, but the first and third one do.
