The
dual of a
planar graph is the planar graph you get by interpreting the graph edges as the borders of areas (including the one 'outer space area', if the graph is finite), using the areas as nodes and connecting the neighbouring nodes with edges.
The Pythagorean solids are examples of such graphs:
For finite graphs, if the graph is connected and only has node of degree at least 3, and the same holds for the dual, then the dual of the dual graph is the original graph.
If not, taking the dual (repeatedly) will lead to this situation.
A planar graph may have multiple different (i.e. non-isomorphic) duals; a given embedding into the plane (i.e. a way to draw it without any crossing lines) the dual is uniquely determined (up to isomorphism).