Let me jump ahead here and tell you a little bit about how a polyhedron and
its dual are related. A polyhedron **P** and its dual **P*** have the
same number of edges, but the roles of the vertices and faces are reversed.
That is, **P*** has as many faces as **P** has vertices, and vice versa.

Also, if **P** is a polyhedron and **P*** is its dual, it turns out
that the dual of **P*** is **P**; that is, **(P*)* = P**. This is
reminiscent of the fact that every (nonzero) number has a unique inverse, and
each is the inverse of the other. In fact, the dual of a polyhdron is
sometimes called the reciprocal polyhedron!

Let's go back now and see how the dual of a polyhedron **P** is derived.
There are three steps:

- Put a point in the center of every face of
**P**; these are the vertices of**P***. - If two faces of
**P**share an edge, connect the center points of those two faces; these are the edges of**P***. - Now just color the faces of the new polyhedron and erase the original.

**Please Note:**

- Step 1 tells us that the number of faces of a polyhedron is equal to the number of vertices of its dual.
- Step 2 tells us that the number of edges of a polyhedron is equal to the number of edges of its dual.

Step 1: Mark face centers | Step 2: Connect adjacent face centers | Step 3: Color the new faces |

Please notice that every face of the Cube has four sides, so four edges of the dual join at each vertex of the dual. Notice also that every vertex of the Cube is surrounded by three faces. When you connect the centers of these faces you get a triangle, so the faces of the dual must all be triangles.

Step 1: Mark face centers | Step 2: Connect adjacent face centers | Step 3: Color the new faces |

Just like the Cube, every vertex of the Dodecahedron is surrounded by three faces, and so the faces of the dual must be all triangles. Likewise, notice that every face of the Dodecahedron has five sides, so every vertex of the dual joins 5 edges.

Cube-Octahedron Compound | Dodecahedron-Icosahedron Compound |

The examples above accounted for four of the 5 Platonic Solids. Recall that the dual of a polyhedron exchanges the roles of vertices and faces. Since the faces and vertices of the Platonic Solids are all identical, it should be no suprise that the dual of a Platonic Solid is another Platonic Solid. As an exercise see if you can figure out what the dual of the Tetrahedron is.

If you wish to check your answer here is a VRML image of the Compound of the Tetrahedron and its dual.

The Archimedean duals are also known as the Catalan polyhedra, after the French mathematician Eugene Catalan, who first described them in 1865.

Duality is defined in terms of polar reciprocation about a given sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x^2 + y^2 + z^2 = r^2, the vertex (x0, y0, z0) is associated with the plane x0*x + y0*y + z0*z = r^2.

Back to the Polyhedral Solids page

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