We usually say that a graph is planar if it can be embedded into 2-space s.t. no edges intersect. Here's a different way to describe the same situation: a graph is planar if it can be embedded into 2-space s.t. the edges form the boundaries of cells which nowhere overlap.
My question: Can you take this second definition and extend the idea to higher dimensions? For instance, allow the cells to be volumes in 3-space - like soap bubbles where the graph edges are the intersection of soap film walls.