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.
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.
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.Formally this is called an involution operation.
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.
Kepler-Poinsot polyhedra:[show]
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Also in the main image are the compounds of the Platonic solids and their duals. Compounds can be created using regular polyhedra which are either the Platonic solids or the
Kepler-Poinsot polyhedra. They are composed together around either an intersphere or a midsphere where their edges intersect. Compounds of regular polyhedra and their duals are interesting because they clearly show the unique relationship between their faces and vertices. There are five of these polyhedra.
A More Mathematical Description
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The dual polyhedron for any polyhedron can be constructed through a process called
reciprocation (also known as
polar reciprocation) which is a kind of transformation about a
concentric sphere that turns vertices of the polyhedron into the
faces of the corresponding dual polyhedron and turns faces of the polyhedron into
the vertices of the dual.
- 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.
Reciprocation about a Circle
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Reciprocating about a circle involves finding a line that is associated with a specific point, or pole, by
inverting that point in relation to a circle and then drawing the line, or polar, that passes through that point of
inversion.
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.
Poles and polars are reciprocal, meaning that when you reciprocate once, poles turn into polars and vice versa, and so when you reciprocate twice you get back to where you started.
Reciprocation about a Sphere
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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.
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The applet above can be seen on its own page at:
3D Reciprocation
Dorman-Luke Construction
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Another way of finding the dual of a polyhedron is by the
Dorman-Luke construction. This construction involves constructing each face of the dual polyhedron by using the vertex figure of the original polyhedron. The vertex figure is the shape that appears when a vertex is cut off of a polyhedron as you can see in the left image below.
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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References
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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.|]]