In every plane graph drawn in the plane Â2 we visually distinguish an external face and internal faces placed inside it. In the octahedron plane graph the external face is {1,2,3}, and other (internal) faces are placed inside it. Having in mind that the plane graph obtained corresponds to an octahedron with equal faces, from the geometrical point of view it is better to try to imagine every plane graph as a projection on a sphere S3

Every planar map M with v vertices, e edges, f faces and c components satisfies Euler's formula

v-e+f = c+1.

The term c = v-e+f obtained from a map on any surface is called the Euler characteristic of the surface. 
 
 

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