The subsets you partition in are necessarily non-measurable sets. That makes them very strange. Any reasonble set that you can come up with is measurable. Even the set of points with rational coordinates is measurable. Of course you cannot partition a gold cube into two pieces, one with the points that have rational coordinates, and the other with points that have irrational coordinates. Non-measurable sets are even weirder.

This is only an indication that arbitrary subsets of R^3 are not a good way to model three dimensional space.