A hex grid is special in 2D geometry because it can tile itself in a way such that any two hexagons that touch a corner must also touch by an edge. This makes it useful for game planning because the distance algorithm does not depend on whether corner movements are allowed or not.
Is there such a shape that's used this way in 3D (possibly based on expanding a sphere-packing like the hexagon), such that any two shapes that touch by a corner or edge must also touch by a face?