I'm talking about a fractal in terms of [Hausdorff](https://en.wikipedia.org/wiki/Hausdorff_dimension) or [Minkowski–Bouligand](https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension) dimension.

Thinking about the Chaos Theory for a while, I have a question about real-world fractals.
I've found appropriate articles about asphalt [Fractal dimension analysis of the fine aggregate gradation of interlocking skeleton asphalt mixture](http://dx.doi.org/10.1007/s11595-011-0269-2) ([RG](https://www.researchgate.net/publication/225724814_Fractal_dimension_analysis_of_the_fine_aggregate_gradation_of_interlocking_skeleton_asphalt_mixture)) and skin [A fractal-like structure in the skin](https://www.worldscientific.com/doi/abs/10.1142/S0218348X96000194).
Also, there is a well-known example with a [coastline](https://en.wikipedia.org/wiki/Fractal_dimension) and [others](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension#Random_and_natural_fractals).

My intuition says that given a quantizable world without quantum effects, there are no fractals. A subquestion: is this intuition right?

Quantum effects make me feel confused.
I've got only one semester of Quantum Physics in university because my major was Mathematics, so I know really little about the quantum world.
The following things make me think that it may be impossible to calculate the Minkowski or Hausdorff dimension of quantum objects (and macro objects, consisting of the quantum ones):

* First, observation of tiny objects like electrons is tricky, so we don't even really know their shape.
* Second, the wave-particle duality.
* Third, the Heisenberg inequality, which may not allow us to calculate coverage of quantum objects.

I can formulate my question in four parts:

* Are there proven fractal real-world objects?
* Are there proven non-fractal real-world objects?
* Are these questions still unsolved?
* Are these questions correct? Is there something (like Heisenberg inequality) that makes these questions incorrect/unsolvable?