Transforming the Cube

This edition of Geometer's Corner will focus on how one can transform a Cube or Hexahedron in a very systematic way into a variety of other cube related solids. We will use GeoGebra 6 3D Graphics to create the polyhedral, and we will be use two types of transformations both known as truncations: Vertex Truncation and Edge Truncation. The cube is one of the Platonic Solids, which are the only convex polyhedra that are vertextransitive (or isogonal), facetransitive (or isohedral) and edge-transitive (or isotoxal) at the same time. This means that each vertex, face, or edge can be moved onto any other by a symmetry operation. In what follows, once we have created a truncated face, we will use rotations about the coordinate axes to build each of these polyhedra; The Truncated Cube, the Cuboctahedron, the Truncated Cuboctahedron and the Rhombicuboctahedron. We will make extensive use of Geo- Gebra's Polygon Tool (Figure 1) to plot polygonal faces, and we will always start with Cube ABCD and face ABCD (Figure 2).