Duality is an important general theme that has manifestations in almost every area of mathematics. Over and over again, it turns out that one can associate with a given mathematical object a related, “dual” object that helps one to understand the properties of the object one started with. Despite the importance of duality in mathematics, there is no single definition that covers all instances of the phenomenon. So let us look at a few examples and at some of the characteristic features that they exhibit.
1 Platonic Solids
Suppose you take a cube, draw points at the centers of each of its six faces, and let those points be the vertices of a new polyhedron. The polyhedron you get will be a regular octahedron. What happens if you ...