I have read that there exist non-integer fractal dimensions and the images generated from these dimensions look organic and they seem to provide a new way of describing the world around us, which raises the question. How many dimensions (spatially or otherwise) do we live in? 3, 3.1, 3.5 as well as 1 time dimension or could time itself exist in a fractal dimension of its own.
Off course the Idea of having fractal dimensions in our real physical word is extremely interesting, but till now, and despite of all the experiments made, there is no evidence of such a thing (for example, in LHC there is a dedicated team of physicists that analyzing colider data for any variation that may be treated as an evidence for extra dimensions).
Thus we can say for now, that within our current technologies and measurement apparatus accuracy, 4 desecrate dimensions are just enough to describe most of what we know about our universe
Anyway there is still many open questions that some physicists argue that they can be solved by considering fractal dimensions, but the most important part in those claims still missing: the experimental evidence, and we mast tend to search for the simplest explanations for those questions, and calming that it's a fractals is a very exotic one.
The world is incredibly well-described locally by $\mathbb R^4$ and we know no experiment that would pretend the contrary.