  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.
