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it is the property of solid crystal that they have straight edges and flat faces.
A cylinder has 0 vertices. If an edge is defined by the meeting of two faces then there are two edges but if an edge is defined by the meeting of two PLANE faces then there are 0 edges.
Uncut diamonds are not any particular shape, though diamond crystals are often octahedra. Cut diamonds usually have a lot more than 8 faces. The most popular, the brilliant cut, has 57 or 58 faces.
It is the set of points, in 3-dimensional space, defined by the intersection of two planes which define faces of the shape.
Faces