This page was inspired by a discussion with some friends. They were looking for a set of polyhedra that were composed entirely of congruent rectangles (other than a cube). The solution came from taking particular polyhedra and replacing opposite faces with prisms that ran through the original polyhedron.
There were two important factors that needed to be considered. First, the opposite faces must be aligned with each other. For example, in an octahedron, opposite triangles are rotated by 180 degrees with respect to each other, and could not be used to constuct prisms. Second, all of the faces must be the same distance apart in order to produce rectangles that were the same height. So a truncated cuboctahedron would not work, even though the opposite faces are aligned, because the octagons, hexagons and squares are at different distances from their respective opposite faces.
In the end, the only figures that met both of these criteria were the rhombic dodecahedron and triacontahedra. Their opposite faces are aligned, and all faces are the same distance from the center.
Below are the original figures, and the rectanglicized polyhedra that were generated from them.
This does not mean that other rectanglicized polyhedra will not be as fun to make or cool to look at, so feel free to explore the other possibilities. If we throw out the need for the rectangles to all be the same height, we can include figures like the Truncated_Cuboctahedron and its respective Rectanglicized Polyhedron.
Figure out what other polyhedra have opposite faces aligned in this fashion, that would make cool rectanglicized polyhedra.