Intuition about the faces in the connected planar graphs

Question

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer (if I have understood it properly); similarly for $K_3$, there are two faces one bounded by edges and the other unbounded regions. In both these cases Euler formula is satisfied.

But when considering the idea of counting faces in Platonic solids, I just don't understand how are we accounting for the outer-face. I guess my instructor explained it with something like mapping into a sphere and back and he says this implies that we can choose to make any face the outer but I didn't understand this approach.

I can see the Euler formula working for (which it should) for the regular polyhedra, but I just can't see how are we taking in account the outer face.

Could anybody help me in understanding these outer-face intuition for these Platonic solids?

Answer 1

First, your typical looking cube graph or whatever it's called. Then, you put the graph on a sphere (use your imagination for the other side), with the gold dot representing the "point at infinity" under stereographic projection. Oh, but let's put it in the plane - where'd the back face go I wonder?

Answer 2

It is not just Platonic solids, but the whole plane. If you take the plane $\mathbb{R}^2$ and add a point at infinity you have a 2-sphere like the surface of the earth. The "far boundary" all connects to the "point at infinity". I think this is a difficult concept-you can try Wikipedia and see if it helps.

Answer 3

Here's a graphical elaboration of what I was talking about in the comments, using the dodecahedron as an example:

(I might consider doing animations for the other polyhedra if asked nicely...)