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A dual is almost like the opposite of a given polytope. For example, a regular octahedron is the dual of a cube.

Look at the similarities between duals with the example of a cube and regular octahedron:

Cube:

Vertices: 8

Edges: 12

Faces: 6

Edges per vertex: 3

Type of face: square (4-sided)

Regular Octahedron:

Vertices: 6

Edges: 12

Faces: 8

Edges per vertex: 4

Type of face: triangle (3-sided)

Both of these shapes can be put together to form a compound and can be rectified to form the same new shape: a cuboctahedron.

Also, duals can fit perfectly inside another where each edge touches the face of the other.

In this same way a regular dodecahedron is the dual of a regular icosahedron. Some polyhedra like the regular tetrahedron is the dual of itself. All polyhedra have duals. Polygons, polychora, and other polytopes can also have duals in a similar fashion.

Q: What is a dual?

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For every polyhedron, there is a dual which is a polyhedron that has:a face where the first had a vertex,a vertex where the first had a face,the same number of edges.A self-dual polyhedron is a polyhedron whose dual is the same shape.All pyramids, for example, are self-dual.

The difference between primal and dual are that primal means an essential, or fundamental of an aspect where as dual means consisting of two parts or elements. Primal is one, dual is two.

One way in which Platonic solids are related is by duality. To construct the dual of a Platonic solid take the vertices of the dual to be the centres of the faces of the original. The lines joining adjacent centres of the original form the edges of the dual. In this way, the numbers of faces and vertices are swapped while the number of edges remain the same.A tetrahedron is its own dual.A hexahedron (cube) and octahedron from a dual pair.A dodecahedron (cube) and icosahedron from a dual pair.One way in which Platonic solids are related is by duality. To construct the dual of a Platonic solid take the vertices of the dual to be the centres of the faces of the original. The lines joining adjacent centres of the original form the edges of the dual. In this way, the numbers of faces and vertices are swapped while the number of edges remain the same.A tetrahedron is its own dual.A hexahedron (cube) and octahedron from a dual pair.A dodecahedron (cube) and icosahedron from a dual pair.One way in which Platonic solids are related is by duality. To construct the dual of a Platonic solid take the vertices of the dual to be the centres of the faces of the original. The lines joining adjacent centres of the original form the edges of the dual. In this way, the numbers of faces and vertices are swapped while the number of edges remain the same.A tetrahedron is its own dual.A hexahedron (cube) and octahedron from a dual pair.A dodecahedron (cube) and icosahedron from a dual pair.One way in which Platonic solids are related is by duality. To construct the dual of a Platonic solid take the vertices of the dual to be the centres of the faces of the original. The lines joining adjacent centres of the original form the edges of the dual. In this way, the numbers of faces and vertices are swapped while the number of edges remain the same.A tetrahedron is its own dual.A hexahedron (cube) and octahedron from a dual pair.A dodecahedron (cube) and icosahedron from a dual pair.

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dual space W* of W can naturally identified with linear functionals

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