Why are the cube and octahedron duals of each other?

Duality is a property in mathematics where the role of two classes of objects can be swapped. Provided this is done consistently, many of the properties such as incidence or intersection in the original structure are true in the swapped structure. In the context of polyhedra, duality is about vertices and faces.

One way to find a dual of a polyhedron is to replace the centre of every face by the vertex of the shape that will become the dual. Whenever two faces of the cube share an edge, join up the centres of these faces to form an edge of the dual. Although it may be somewhat difficult to visualise, the dual of the cube is a regular octahedron.

A cube has 8 vertices, 6 faces [and 12 edges]. There are 3 faces meeting at each vertex, each face has 4 vertices.

An octahedron has 8 faces, 6 vertices [and 12 edges]. There are 3 vertices on each face, there are 4 faces meeting at each vertex.