If we examine the number of vertices per face and the number of faces meeting at a vertex in the polyhedra, we can see some of the reasons for duality:
Column A: number of faces meeting at a vertex. Column B: number of vertices surrounding a face. Polyhedron A B ----------------------------- tetrahedron 3 3 cube 3 4 octahedron 4 3 dodecahedron 3 5 icosahedron 5 3
Notice that in each pair, by reversing the two numbers, we can generate the complementary member of the pair. A cube [3,4] is the reverse of an octahedron [4,3]. The tetrahedron [3,3] is self-dual because its dual (the reverse of these two numbers) is itself [3,3].
This numbering system was invented by W.A. Wythoff, and accordingly the set {a,b}, where the numbers a and b represent the information in the columns above for a given polyhedron is known as that polyhedron's Wythoff number.
Here we need to examine the number of vertices in the cells that make up the boundary of the polytopes and the number of those cells meeting at a vertex. Wythoff's technique works fine as long as you're willing to step everything up a dimension and consider cells instead of faces.
Column A: number of cells meeting at a vertex. Column B: number of vertices surrounding a cell. Polytope A B ----------------------------------- 5-cell (4-simplex) 4 4 hyper-cube (8-cell) 4 8 h-cube dual (16-cell) 8 4 120-cell 12 20 600-cell 20 12 24-cell (self-dual!) 6 6
The 5-cell is the four-space equivalent of the tetrahedron--it is self-dual, because the number of cells meeting at a vertex is equal to the number of vertices per cell. What is interesting about this collection is that the 24-cell is also self-dual. It's pretty easy to demonstrate (though we won't here) that every simplex is self-dual under this conception, but it is very unusual (in fact, unique) to find another shape with this self-dual property.
To read more about Wythoff numbers, try here.