|Duals of Platonic Solids|
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.
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.
In our case of dual polyhedra, vertices of polyhedron correspond to faces of the dual polyhedron and vice versa. Duality in polyhedra also has the interesting property that if you find the dual of a polyhedron and then find the dual of the dual you end up back with your original polyhedron.
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
Reciprocation about a Circle
Reciprocation about a Sphere
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.
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- 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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[[Description::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.|]]