There exist families of four polyhedra each with interesting internal relationships.
In the following illustrations the colors of the arrows have a special meaning.
white arrow means: the face centers transform to corners,
gray arrow means: the middle of the edges transform to corners, and
black arrow means: the middle of the edges transform to face centers.
The white arrows appear in pairs and are pointing in both directions. This reflects the wellknown duality between to polyhedra.
The transformation from center of edge to centre of face is achieved by putting planes in each center of edge which are normal to the vector pointing to the center of the polyhedra. This transformation is equivalent to a special stellation (star forming) where the pyramids on each face reaches exactly the height which lets disappear the old edges. (Würfel = cube)