  There are two chiral Archimedean polyhedra, the snub cube and snub
dodecahedron together with their duals the Catalan solids, pentagonal
icositetrahedron and pentagonal hexacontahedron. In this paper we construct the
chiral polyhedra and their dual solids in a systematic way. We use the proper
rotational subgroups of the Coxeter groups $W(A_1 \oplus A_1 \oplus A_1)$,
$W(A_3)$, $W(B_3)$ and $W(H_3)$ to derive the orbits representing the solids of
interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and
snub dodecahedron respectively. We prove that the tetrahedron and icosahedron
can be transformed to their mirror images by the proper rotational octahedral
group $\frac{W(B_3)}{C_2}$ so they are not classified in the class of chiral
polyhedra. It is noted that the snub cube and the snub dodecahedron can be
derived from the vectors, which are non-linear combinations of the simple
roots, by the actions of the proper rotation groups $\frac{W(B_3)}{C_2}$ and
$\frac{W(H_3)}{C_2}$ respectively. Their duals are constructed as the unions of
three orbits of the groups of concern. We also construct the polyhedra,
quasiregular in general, by combining chiral polyhedra with their mirror
images. As a by product we obtain the pyritohedral group as the subgroup the
Coxeter group $W(H_3)$ and discuss the constructions of pyritohedrons. We
employ a method which describes the Coxeter groups and their orbits in terms of
quaternions.
