If I have a graph consisting of 2 disjoint triangles, which are connected by an additional edge, then I have difficulties understanding how its dual graph looks like.
The dual graph consist of three vertices 1, 2 and 3 such that there are three edges between 1 and 2, three edges between 2 and 3, one loop at the vertex 2.
Note that the number of faces of the graph equal the number of vertices in the dual graph, number of edges equal the number of edges and the number of faces equal the number of vertices of the dual graph.
Further a planar connected graph is isomorphic to its double dual.
The following image is the best I could get with MS Paint. Sorry to have it hoorible!