The Dual of a Polyhedron

Regular Patterns, The Dual of a Polyhedron

The Dual of a Polyhedron

When a polyhedron is mapped onto a sphere, it looks like a map of countries. We can replace each country with a dot and draw edges between the dots that represent neighboring countries. A ring of dots and edges surrounds what use to be a vertex. Thus corners have become faces and faces have become corners. The new edges are perpendicular to the old edges, hence there are just as many edges as before. With v and f swapped, v-e+f is still 2, as expected. This process creates the dual of a polyhedron. It is called the dual because the dual of the dual resurrects the original graph. Let's illustrate with the octahedron.

Stand the octahedron up on one corner as we did before. Four faces are turned upward to the sky and for faces are slanted down. Now place a dot in the center of each face. Connect the four dots on top in a square, since those four faces form a ring. Similarly connect the bottom four dots in a square. Finally connect each top dot to its dot below. You have drawn a cube. Each face of the cube contains a vertex from the octahedron, just as each corner of the cube sits in the middle of a face of the octahedron. As an exercise, draw the dual of the cube, placing a vertex in the middle of each square face, and get the octahedron back again.

The dodecahedron and icosahedron are dual, and the tetrahedron is its own dual.