Question

Here is a picture of an octahedron, which is a regular (Platonic) solid with 8 triangular faces, 12 edges, and 6 vertices. You can imagine an octahedron as two pyramids with square bases, which are then glued together along their bases. octahedron We can turn a polyhedron into a graph by placing its vertices in the plane, and adding edges between those vertices which share an edge on the solid. If we do this to the octahedron, we get the graph below.

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Graph For Octahedron: For dual graph, we need to replace faces with vertices and in dual edge between these vertices correspond to edges separating the faces in graph. This dual graph has 6 faces, 12 edges and 8 vertices. Which is same as the first graph so Dual of Dual a graph is itself. \r\n icosahedrons rightarrow^dual dodecahedron As per the procedure faces rightarrow^dual vertices icosa rightarrow^dual dodeca edge rightarrow^dual edges (20f, 30e, 12v) rightarrow (20v, 30e, 12f) vertices rightarrow^dual faces

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