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Awesome and very elegant proof especially as we know that all closed convex surgaces (n-gon's) must satisfy Eulers equation.

My math skills aren't what they used to be, so instead of using calculus, I cheat. I tend to just imagine a cylinder as a rectangle when when its curved face is cut & flattened. Then consider it to be one face and where the two side edges of that rectanglular face met as a third edge, and where that 3rd edge intersects the 2 top & bottom circular edges as 2 vertices. Doing so, Euler's formula is satisfied. V-E+F=2-3+3=2.

For the closed cone if cut down the face perpendicular to the bottom edge, it flattens out to an isosceles triangle so again one extra edge where those two sides meet and a verticie at each end. Si with only 2 faces, 2 verifies and 2 edges again Eulers equation is satisfied.

For the open cone, that loses a bottom face. You have to count the inner face instead.

For the sphere I realize you make one by rotating a semi circle around an axis 360°. So I consider the arc that they meet at 0° & 360° (or the axis of rotation) as an edge, with the two poles at it's endpoints. So with one face, one edge and 2 verticies, again Eulers equation holds.

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