Duality for platonic solids

This is a tetrahedron

If we mark the centre of each square face of the tetrahedron, and join these points together, we get another tetrahedron, pointing the opposite way.

If we mark the centre of each face of the smaller tetrahedron, and join these points together, we get an even smaller tetrahedron, with the same orientation as the original one. In other words, the tetrahedron is its own dual.

This is a cube

If we mark the centre of each square face of the cube, and join these points together, we get an octahedron. In other words, the dual of a cube is an octahedron.

If we mark the centre of each face of the octahedron, and join these points together, we get a cube, which is just a smaller copy of the solid that we started with. In other words, the dual of an octahedron is a cube.

This is a dodecahedron

If we mark the centre of each pentagonal face of the dodecahedron, and join these points together, we get an icosahedron. In other words, the dual of a dodecahedron is an icosahedron.

If we mark the centre of each face of the icosahedron, and join these points together, we get a dodecahedron, which is just a smaller copy of the solid that we started with. In other words, the dual of a an icosahedron is a dodecahedron.