[gohrt]

(I think) this is a case were lay intuition assumes objects are piece-wise differentiable, but the formal math definitions don't.

That is, most people think of physical objects, where there is a minimum "planck length", and no fractal-ish structures with infinitely fine-scale structure.

It is still abstractly interesting that even 2-D fractals satisfy the Jordan Curve therorem, whie 3-D faractals do not satisfy the 3-D version of the conjecture.