Branching of the W(H ₄) polytopes and their dual polytopes under the coxeter groups W(A ₄) and W(H ₃) represented by quaternions

4-dimensional H4 polytopes and their dual polytopes have been constructed as the orbits of the Coxeter- Weyl group W(H ₄) , where the group elements and the vertices of the polytopes are represented by quaternions. Projection of an arbitrary W(H ₄) orbit into three dimensions is made preserving the icosahedral subgroup W(H ₃) and the tetrahedral subgroup W(A ₃) . The latter follows a branching under the Coxeter group W(A ₄) . The dual polytopes of the semi-regular and quasi-regular H ₄ polytopes have been constructed.