Starting at some point in an infinite 2D grid, there is a simple and intuitive path to visit every other point exactly once using only unit-length movements in the directions of the basis vectors. It's a discrete spiral that looks like this:
.----.----.----.----.
| |
. .----.----. .
| | | |
. . .----. .
| | |
. .----.----.----.
|
.----.----.----.----.----.
The infinite continuation of this pattern is guaranteed to visit all points $(x,y)$ $\vert$ $x,y \in \mathbb{Z}$. Obviously, such a pattern cannot exist in 1D, since movement in one direction leaves all points in the opposite direction inaccessible without visiting some point twice.
I want to know if there is a proof for or against the existence of such a path in 3D. I naively tried to trace a path around a 3x3 cube but was unsuccessful. I tried again with a 2x2 cube and the 4x4 cube enveloping it, but this failed as well. After a few other failed leads, I gave up on trying to trace the path out by hand. I was, however, able to find a path under such constraints in 4D.
I am not looking for a Hilbert curve or any other space-filling curve that maps the unit line segment to the unit cube. I'm specifically looking for a path that, starting at some point and moving only one unit at a time along the x, y, or z axis, is guaranteed to visit every point $(x, y, z)$ in an infinite 3D grid exactly once.
