Apropos of the question "Why is the valency 4 on average?" (In Warren's argument), here's a mental picture of the generic situation: An edge in the graph is the intersection of the oblique plane with a face of a cube in the tessellation. A vertex A vertex in the graph is the intersection of the oblique plane with an edge of a cube in the tessellation. Four faces surround each edge in the tessellation, so four edges surround each vertex in the graph. Jim Propp