I have a question regarding my dissertation I could not find an answer to:
Can a 3D-Curve (whose graph goes in all three directions of space) have a Fractal Dimension (FD) anywhere between $1$ and $3$, depending on its space filling capability?
The context of my question is as follows: a space filling curve in 2D-Space has a FD of $2$. A space filling curve in 3D-Space has a FD of $3$, e.g a 3D Hilbert Curve.
A non-space filling curve that lies in 2D-Space (e.g a 2D Koch Curve) should have a FD between $1$ and $2$, depending on its complexity/ space filling capability. And such a curve that in 3D-Space should have a FD between $1$ and $3$?
N.B at first I thought that a curve in 3D-Space should have a FD between 2 and 3, e.g the 3D-Sierpinski Arrowhead Curve, which should have the same FD of $2.32$ as the 3D Sierpinski Gasket. However I came to the conclusion that a 3D-Curve could be simple enough to have a FD below $2$ (also backed by box-counting measurments I made). A 3D Curve which my 3D Box-Counting Algorithm says has a FD of $1.52$.
Regards