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polyhedrons need flat face and edges, corners which cylinder cones don't have.
Two polyhedrons have 18 edges: truncated tetrahedron and hexagonal prism.
I can just think of the pentagonal pyramid.
Yes. to add to that a vertex must be connected to at least 3 edges to be 3d, an edge is always connected to 2 vertexes, so the closest the two can ever be is vetexes x 3 = edges x 2, but when working with any platonic solid you can follow this: vertexes x (faces / vertexes) x [edges on one side] = edges x 2 or vertexes x [faces meeting at one vertex] = edges x 2 when working with any other polyhedron [vertexes with x amount of faces] x (x) + [vertexes with y amount of faces] x (y) ...{and so on} = edges x 2
For a simply connected polyhedra, the Euler characteristic requires that E + 2 = F + V