Is there a dimension which extends to negative or even irrational numbers? Just elaborating on the question:
We all use to natural numbers as dimensions: 1 stands for a length, 2 for area, 3 for volume and so on. Hausdorff–Besicovitch extends dimensions to any positive real number.
So my question: is there any dimension which extends this notion further (negative numbers or even irrational numbers)? If so, what are the examples, how it can be used?

The infinite lattice is a fractal of negative dimension: if you scale the infinite lattice on a line 2x, it becomes 2x less dense, thus 2 scaled lattices compose one non-scaled. If you take a lattice or on a plane, scaling 2x makes it 4x less dense so that 4 scaled lattices compose one non-scaled, etc.
