I would like to know is it possible to generate a fractal in the plane with dimension higher than 2? If that is possible, please could you explain the intuition behind that? If it is not possible, is there some proof for that? Thank you in advance.
Best regards,
Meaning of "dimension"? If you mean Hausdorff dimension, then NO. If $A \subset B$, then $\dim A \le \dim B$. And $\dim \mathbb R^2 = 2$.