In this paper, we present a method to create a new class of polyhedra. All the faces of these polyhedra are bounded by smooth (quadratic B-spline) curves and the face boundaries are C1 discontinuous everywhere. These polyhedral shapes are limit surfaces of a generalized vertex truncation subdivision scheme. We obtain an approximation of these smooth and fractal polyhedra by iteratively applying a new vertex truncation scheme to an initial manifold mesh. Our vertex truncation scheme is based on Chaikin's construction. If the initial manifold mesh is a polyhedra only with planar faces and 3-valent vertices, in each iteration we construct a polyhedral mesh in which all faces are planar and every vertex is 3-valent,