This is a set of random ideas that just want to jump out of my head and onto the internet. Who knows whether office chair philosophy is any more credible than arm chair philosophy, but you decide, and let me know.
I just discovered that shapes with half-integer dimensions have a simple square construction by subdividing by 4. An example is labelled the quadratic type 2 curve (or sometimes Minkowski sausage) with dimension 1.5. However this curve comes in a left and right handed form. Another example I made here:
It is also possible to build 2.5D surfaces, an example is here, however this surface intersects. It is possible to make a non-intersecting surface, which is approximately a 3D version of the above:
And here's a 1.5D curve in 3D space:
A 2.5D curve in 3D space is a bit more of a challenge