Is there a space filling "tube"? I am trying to construct a curve that fills the space as much as possible while maintaining certain minimum distance between two segments of the curve. Intuitively, this should be some sort of space filling "tube".
I could wrap a boundary around a space filling curve, e.g. Hilbert curve. However this certainly is not an optimal solution as the allowed maximum width of such "tube" is not continuous when the order of the curve increases.
How can I construct such a tube which fills the space as much as possible?  
 A: Let me address filling a large box in $\mathbb{R}^3$ rather than filling all of $\mathbb{R}^n$.
Here is one idea, taken from the MO quesiton, "Coiling Rope in a Box."
Use one of the two patterns below to fill one layer, and then repeat
for the next layer above, and so on. (The two patterns achieve the same density.)
Connect adjacent layers by a short vertical section.



A: You should be able to realize the density of the optimal planar disk packing in the limit.
Denote the diameter of the rope by $\rho$. The basic module is a rectangular $a\times b\times L$ box filled with parallel strands  of length $L-2\rho$. The strands are arranged such that a normal cross section of the box shows  a densest  packing of circles of diameter $\rho$. The strands  are spliced together in two $a\times b$ layers of thickness $\rho$ at the ends of the box, in order to get a single long piece of rope within the box. 
Now fill space with such boxes in a clever way (making sure that the rope stays connected), such that the boxes get ever larger as you go out. In this way you can make the "overhead" disappear in the limit.
