See, we can divide graphs into planar ones and non-planar ones. This seems to make a lot of sense until you find that this seems to only work in two dimensions.
I can't think of any graph that cannot have a projection in three-dimensional space without any lines intersecting. Obviously, there are non-planar (in the 2d sense) graphs, such as the Petersen graph, but AFAIK it is very easy to make a 3D Petersen graph without any lines intersecting.
What is so special about 2 dimensions that allows nontrivial planar and non-planar graphs? In one dimension all nontrivial graphs are non-"linear", and it seems all three-dimensional graphs are "spacear".