Necropost, but I try to fill in holes of misunderstanding whereever they might be stumbled into.

"Fractal dimension" is not the same as "Fractional dimension". A fractal's dimension relates to how intricate and complex its surface iteration is. It's basically how much larger the surfaces gets per amount you increase its resolution. It nothing directly to do with spacial dimensions like length, height, or width.

HOWEVER, there is a INDIRECT correlation between them via topological flattening. The idea is that an objects dimensionality is the smallest number of dimensions needed to fit the points. Example: A crumpled sheet of paper (platonic ideal) exists in 3D, but you could flatten it out to 2D, so it's a 2D object in 3D space. You couldn't flatten it down to 1D.

A fractal outline (irrational or repeating) is "too long" to fit in 1D, despite being a single uninterrupted line. The closer you look, the longer it is. You wouldn't need two whole dimensions to fit it though. Fractal dimension.

So fractional spatial dimensions are still theoretically nonsensical, but you could (mathematically) describe a 4D object that can just barely fit in 3 1/3 dimensions.