4D-Polytopes and Their Dual Polytopes of the Coxeter Group $W(A_4)$ Represented by Quaternions

4-dimensional $A_4$ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(A_4)$ where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary $W(A_4)$ orbit into three dimensions is made using the subgroup $W(A_3)$. A generalization of the Catalan solids for 3D polyhedra has been developed and dual polytopes of the uniform $A_4$ polytopes have been constructed.