As I detail at length in the book Basic Platonic and Archimedean Solids, there are many relationships between three of the Platonic solids: cube, tetrahedron, and octahedron. You can start with any of these shapes and perform a few simple "tricks" to get the other shapes.
Yesterday I created a new video to demonstrate one of these tricks: how to create a cube and use it to create tetrahedrons and an octahedron.
The detailed, written version of this project is included in April's Projects of the Month.
Anyone can design anything in 3D! http://www.3dvinci.net/