Dual Polyhedron
From Math Images
| Duals of Platonic Solids |
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Duals of Platonic Solids
- This image shows the five Platonic solids in the first row, their duals directly below them in the second row, and the compounds of the Platonic solids and their duals in the third row.
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Basic Description
All polyhedra have a corresponding dual polyhedron. This means that if you start with one polyhedron and transform it in a specific way you are able to find another polyhedron that is the unique counterpart of your original polyhedron.
In mathematics duality is a complex term that has many uses and definitions. In general duality refers to a property that allows you to change a mathematical idea, theorem, statement, concept, etc... into something that corresponds with what you started with. For example, there is a dual relationship between family members. If we state: Alan is the father of Tanya. Then our corresponding dual statement would be: Tanya is the daughter of Alan.
As you can see in the main image and will learn more about later, the dual of the cube is the octahedron and vice versa. The figure to the left illustrates these unique properties between the faces and vertices of the cube and octahedron.
So for example, we can state: A cube has 8 vertices and 6 faces. Because of duality, we are able to make another true statement about the dual of a cube just by switching the terms 'vertex' and 'face'. This new dual statement is of course: An octahedron has 8 faces and 6 vertices. Another statement might be: each face of a cube has 4 vertices and each vertex has 3 faces. The dual statement would be: each vertex of an octahedron has 4 faces and each face has 3 vertices.
In general it is complicated to construct the dual of a polyhedron but we find a special case in Platonic solids. Every Platonic solid has a corresponding dual polyhedron that can be found through a fairly easy process. The simplest way to create the dual polyhedron for a Platonic solid is by finding the midpoints of each of the faces, and then connecting these midpoints so that they become the vertices of the new dual polyhedon. Take another look at the picture with the octahedron and the cube. You can see exactly how this method works with Platonic solids.
As you can see in the main image above, the dual polyhedra of the Platonic solids are all Platonic solids themselves. So, the cube and the octahedron are duals of each other; the dodecahedron and the icosahedron are duals of each other; and the tetrahedron is the dual of itself. In other words, the tetrahedron is self-dual. Dual polyhedra exist for all other polyhedra as well; see "A More Mathematical Explanation" below.
A More Mathematical Description
- Important facts about dual polyhedra:
- The dual polyhedron of a dual polyhedron is the original polyhedron.
- If a polyhedron has E edges, F faces, and V vertices, its dual has E edges, V faces, and F vertices.
- Important facts about dual polyhedra:
Reciprocation about a Circle
More specifically, the polar of a point in a circle is the line that goes through its inversion point and is also perpendicular to the line containing the original point and the center of the circle. The figure to the right illustrates this process of reciprocation. The point is labeled B; the inversion point is C; and the center of the circle is A. The polar of point B is the line that passes through the point, C.
The pole of a line with respect to a circle is the inverse of the point on the line that is closest to the center of the circle. Notice this again in the figure to the right. The pole of the line through C is the point, B.
If the point lies on the circle then its polar is the line that is tangential to the circle at that point.
Reciprocation about a Sphere
After understanding the process of reciprocation in plane geometry, it is then fairly easy to generalize how reciprocation will work in higher dimensions. In our case (polyhedra in 3 dimensions), we just need to figure out reciprocation using a sphere as opposed to a circle. This process of reciprocation about a sphere is what is necessary to create a dual polyhedron. Reciprocating about a circle in two dimensions allows you to transform a poly"gon" into a corresponding dual poly"gon".
Begin with any sphere that is concentric with the polyhedron. This sphere will be analogous to the circle we used in the previous section. Changing the radius of the concentric sphere will change the size of the associated dual polyhedron, but all of the duals, regardless of size, will be similar. Instead of constructing a polar line associated with each vertex of a polygon, we construct a polar plane associated with each vertex of a polyhedron using the same process of inversion. The plane is constructed perpendicular to the ray that goes from the center of the sphere through the vertex.
These planes are then the faces of the dual polyhedron we had wished to construct. Where these faces intersect are the edges of the dual polyhedron, and where the edges intersect are the vertices of the dual. The vertices of the dual polyhedron can also be found by the process of reciprocation because they are the poles that are reciprocal to the face planes of the original polyhedron.
In other words, the faces of a polyhedron and the vertices of the dual polyhedron are reciprocals. Again because of the property of duality in polyhedra, swapping "face" with "vertex" gives another corresponding true statement: The vertices of a polyhedron and the faces of the dual polyhedron are reciprocals. All of this can be seen in the applet below showing how this process of reciprocation turns polyhedra into their duals.
The applet above can be seen on its own page at: 3D Reciprocation
Dorman-Luke Construction
After finding the vertex figure, construct the circumcircle around the it at each of its vertices. Notice the dotted-line circumcircle around the vertiex figure in the image to the right below.
Next construct a line segment at each vertex that is tangent to the circumcircle, and connect each of the line segments. This new polygon is a face of the dual polyhedron. The tangent line segments that form the new vertex figure can also be seen in the image to the right below.
Repeating this process for each of the vertices of the original polyhedron and then connecting the constructed faces will create the dual polyhedron.
 Image by: MathWorld |
 Image by: MathWorld |
References
- Wolfram MathWorld, Dual Polyhedron
- Wikipedia, Dual Polyhedron
- New Caledonia, AMC Duality
Future Directions for this Page
More images maybe. Especially for the Reciprocation about a Sphere section
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