The dual of a cube

Instructions

[You may have to press the spacebar or the enter key to start]
Use a right click (or Control-click on a Mac) to change the view angle by dragging the plane in any direction.

Change the size or orientation of the cube by dragging a vertex with a left mouse.

Notice the polyhedron formed by joining the mid-points of each face of the cube is a regular octahedron. This is the 'dual' of the cube.

What would happen if you started with an octahedron instead of a cube?