Dual of a Polyhedron

For every polyhedron there is another polyhedron that is intimately connected to it, that is uniquely defined by it. This related polyhedron is called the Dual Polyhedron.

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:

  1. Put a point in the center of every face of P; these are the vertices of P*.
  2. If two faces of P share an edge, connect the center points of those two faces; these are the edges of P*.
  3. Now just color the faces of the new polyhedron and erase the original.
That's all there is to it!

Please Note:


Example 1 - The Dual of the Cube

As an example, the following sequences of images shows the derivation of the dual of the Cube (you will notice that in the center image, Three of the faces of the cube have been removed so you can see inside).

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.


Example 2 - The Dual of the Dodecahedron

As a second example, the following sequences of images shows the derivation of the dual of the Dodecahedron (you will notice that in the center image, one of the faces of the Dodecahedron has been removed so you can see inside).

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.


Compounds of Dual-Pairs

You now know that the Cube and Octahedron are duals of each other, and that the Dodecahedron and Icosahedron form a dual-pair. The next two images combine these dual-pairs in a way that dramatically illustrates their relationship as duals. Notice that every face has a vertex of the dual above it, so the 1-to-1 relationship between faces and vertices is clear. Also, notice that the edges of one solid cross the corresponding edges of its dual, leaving no doubt that they have exactly the same number of 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.

Random Notes about Duals

A polyhedron and its dual have the same axes of symmetry.

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.


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