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Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions

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.