Regular and Irregular Chiral polyhedra with Coxeter Diagrams and Quaternions

Chiral polyhedra, snub cube and snub dodecahedron are well known regular solids described with respective proper octahedral and proper icosahedral symmetries. In this project we construct the elements of the chiral tetrahedral, octahedral and icosahedral symmetries with quaternions by using the Coxeter-Dynkin diagrams A3, B3 and H3. Chiral group elements acting on a general vector expressed as a linear combination of the weight vectors in terms of two free parameters x and y lead, in general, to the vertices of the solids of irregular polyhedra. It has been proved that for special values of the parameters regular chiral polyhedra can be obtained; regular and irregular chiral polyhedra possess the same symmetry. Each chiral group has been worked out in detail. Dynkin diagram symmetry in the case of A3 diagram extends the chiral tetrahedral group to the pyritohedral symmetry. Duals of the pseudo icosahedra with the tetrahedral and pyritohedral symmetries lead to the tetartoid and pyritohedron solids which correspond certain crystals in nature. It is shown that the centers of the vertices of the pseudo icosahedron describe the pseudo icosidodecahedron possessing irregular pentagons and scalene triangles as faces. Irregular snub cube is a convex solid covered with square, equilateral as well as scalene triangular faces and irregular snub dodecahedron consists of regular pentagons together with equilateral as well as scalene triangles. Regular snub cube and snub dodecahedron are obtained corresponding to the special values of the parameters of x and y.