I don't come from a pure mathematics background, so bear with me. I was thinking about the movie/tv show Stargate, with this picture in particular: James Spader drawing something like what I'm talking about.. It represents how the Stargate uses 6 coordinates based on constellations to uniquely determine a point in space (obviously there is so, so much wrong with the in-universe explanation but I'm ignoring that). I got to thinking: Can Earth's address (as depicted in the show) even cover 6 sides of a cube? So here's a more generalized and abstract version of the question:
Suppose you have a point which is the center of a cube of side length 2. You also have six arbitrary direction vectors, which means you have six points at which a vector intersects the cube's surface. How can we find out whether there exists a rotation that places all 6 points at unique faces? Additionally, how can we find out the maximum (or minimum) number of unique faces which can contain these points?
There obviously exist cases where the vectors each intersect a different face. Trivially, selecting vectors that pass through points (1,0,0) (0,1,0) (0,0,1) and their negatives (as in the picture) accomplishes this.
There also exist cases where the 6 vectors cannot be made to intersect all sides. For example, if you put them close enough together, there ends up always being at least one extra side. My hunch says that the condition for this is if all the angles between them are $ \theta _{ij}< 2cos^{-1} (\frac{1}{3})$, but I don't know if it's correct or how to prove that, I don't know whether that's the necessary and sufficient condition. I also don't know how to choose vectors such that they can cover at most 3 or 4 sides, if that's possible. If you have 3 vectors, you can always just rotate the cube so that there's a corner between them, so I'm pretty sure you can always make 3 or more vectors intersect with at least 3 different sides. Again, though, I'm not 100% sure about that.