bijection between planar graphs and maps

Planar graphs are graphs whose edges do not cross
For this picture I'm imagining maps that are "nice" looking. Like the map of the United States.

1. Injective

Take a map. You can represent it in list form by writing out its regions and the regions that share a border.
The mapping from maps to graphs works like this: put a vertex in the middle of each region. Draw a hashmark at each border, normal to the border. On either side of this border, each hashmark can be extended to reach a vertex without leaving the region. This results in a planar graph.
Changing which regions border what will change the list representation, which will result in a different adjacency matrix for the graph. So the mapping is injective.

Each planar graph has a corresponding planar map. To see this, imagine printing out your planar graph in very thick ink. Then, each edge can be seen, not as a line, but as a portion of a region.