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Google is saying 2 and 3.

What's the real real answer?

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2 Answers

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The notion of a "face" only applies to polytopes, that is, volumes where each of the boundary surfaces are locally linear. Thus a cube has six faces, but a talking about the faces of a sphere is meaningless. And the treatment of the curved boundary of the cylinder similarly has nothing to do with faces.

If by "face" you extend the definition to "any maximal subset of the boundary such that any two points in the subsurface can be joined by a differentiable curve lying within the subsurface" then the cylinder would be considered to have $3$ such "faces."

CHopping one of the boundarys so that it can be laid flat,k as Google does, is kind of a cheat, but it will usually give the same result as that extended definition.

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Where is Google saying that a cylinder has 2 faces? It doesn't say this in the screenshot, although it should because this is the most useful answer - as the following argument shows.

Take Euler's formula $F+V=E+2$, consider the figure produced by drawing a line down the cylinder, like the seam on a tin. Then there are three faces (two 1-sided round ones and one 3-sided cylindrical one), two vertices (where the seam meets the top and bottom rim), and three edges (rim, seam, rim).

If you remove the seam then you reduce the number of edges by 1 by direct deletion and by 2 by merging because a vertex is removed (an edge goes from vertex to vertex: no vertex means no edge), the number of vertices by 2, and you merge the cylindrical faces by removing the seam, thus reducing their number so that instead of 1 of them you have 0 of them. Thus $F+V=2+0=0+2=E+2$.

The intuitive justification for this way of counting is that an $n$-gon is surrounded by $n$ lines (a 1-gon is surrounded by 1 line) but the cylindrical face is not surrounded at all - you can go along it for ever and ever and never hit a boundary. This unusual character is fittingly recognised by calling it a $0$-gon.

Google's quoted argument is defective because it counts the faces not of a cylinder but of a cylinder plus a seam: a different figure altogether.

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