A point where three (or more) faces (sides) meet.
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A cone !
A cone.
An octahedron is a closed 3-d shape with 8 polygonal faces. There are 257 topologically different convex octahedra. The number of faces at a vertex ranges from 3 to 7.
A cone
The concept of a vertex, which is the intersecting point between two or more geometrical shapes, is meaningless in this case. A point can't have a vertex since it has no shape. A vertex, however can be a point. For example, if one 2-D line intersects another 2-D line, that point of intersection is the vertex.