  4-dimensional H4 polytopes and their dual polytopes have been constructed as
the orbits of the Coxeter-Weyl group W(H4) where the group elements and the
vertices of the polytopes are represented by quaternions. Projection of an
arbitrary W(H4) orbit into three dimensions is made preserving the icosahedral
subgroup W(H3) and the tetrahedral subgroup W(A3), the latter follows a
branching under the Coxeter group W(A4) . The dual polytopes of the
semi-regular and quasi-regular H4 polytopes have been constructed.
