  Catalan Solids are the duals of the Archimedean solids, vertices of which can
be obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots
can be represented by quaternions. The respective Weyl groups W(A3), W(B3) and
W(H3) acting on the highest weights generate the orbits corresponding to the
solids possessing these symmetries. Vertices of the Platonic and Archimedean
solids result as the orbits derived from fundamental weights. The Platonic
solids are dual to each others however duals of the Archimedean solids are the
Catalan solids whose vertices can be written as the union of the orbits, up to
some scale factors, obtained by applying the above Weyl groups on the
fundamental highest weights (100), (010), (001) for each diagram. The faces are
represented by the orbits derived from the weights (010), (110), (101), (011)
and (111) which correspond to the vertices of the Archimedean solids.
Representations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions
simplify the calculations with no reference to the computer calculations.
