Catalan solids derived from three-dimensional-root systems and quaternions

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams $A 3$⁠, $B 3$⁠, and $H 3$ whose simple roots can be represented by quaternions. The respective Weyl groups $W ( A 3 )$⁠, $W ( B 3 )$⁠, and $W ( H 3 )$ acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the 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), and (011) 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 $W ( A 3 )$⁠, $W ( B 3 )$⁠, and $W ( H 3 )$ by the quaternions simplify the calculations with no reference to the computer calculations.