Summary

In this thesis, we use polyhedral to construct crystal. And we note almost generate polyhedral of crystal is dual to the convex hull of the crystal.

The exception are Truncated tetrahedron、Truncated cube   and Small rhombicuboctahedron. It is also easy to generate the dual from this three polyhedral, we just connect outmost point of each Rhombic Polyhedra (the green line). Then we got the dual of Truncated tetrahedron.

But for Truncated cube  , if we connect outmost point of each Rhombic Polyhedra, we will obtain Great dodecahedron, it's a Kepler–Poinsot polyhedron. To obtain the dual of  Truncated cube, we must do reflection for each face.

To obtain the Small rhombicuboctahedron, we must construct plane P such that P contain the outmost point of 3 adjacent 12-face Rhombic Polyhedra, and construct a strain L through the center and the outmost point of the 6-face Rhombic Polyhedra, then the intersection point A of L and P is a vertex of the dual of Small rhombicuboctahedron.

Cabri 3D:

Hence we know that at most time we can construct dual of Polyhedra by create crystal, the main reason is the Rhombic Polyhedra is symmetric and the face of generate polyhedral is regular, so the outmost point of Rhombic Polyhedra will project on the center of the face of Rhombic Polyhedra. It is just the definition of dual.