Dual polyhedron

In geometry, every polyhedron has a special partner called its dual. This means that the points, or vertices, of one shape match up with the flat surfaces, or faces, of the other, and the lines connecting points, or edges, also match up in a matching way. While these dual shapes are often studied in a more abstract way, some can actually be built as real 3D shapes.

An interesting fact is that if you take the dual of a dual shape, you get back to the original shape. This special relationship helps us understand how shapes are connected. Duality also keeps the symmetries of a shape the same, meaning that shapes with certain patterns of symmetry have duals that share similar patterns.

For example, the beautiful and well-known Platonic solids, like the tetrahedron, cube, and dodecahedron, each have matching dual shapes. The regular tetrahedron is even its own dual, making it a special case. This idea of duality helps mathematicians and scientists study and understand the hidden relationships between different geometric shapes.

Kinds of duality

There are different ways to think about dual polyhedra. One important way is called polar reciprocation. This method uses a sphere to create a dual polyhedron. Each point (vertex) in the original shape matches a flat surface (face) in the dual, and each flat surface in the original matches a point in the dual.

Another way is topological duality. Even if two shapes can't be made from each other using a sphere, they can still be duals if their points, lines, and flat surfaces match up in a specific way. This idea works for many kinds of shapes, even ones that are not perfectly symmetrical.

Self-dual polyhedra

A polyhedron is called self-dual if its dual looks exactly the same in terms of how its points, lines, and surfaces connect. For example, the dual of a regular tetrahedron is another regular tetrahedron. Every flat shape, like a pentagon or a hexagon, is also self-dual because it has the same number of points as lines.

There are many self-dual polyhedra. Simple examples include pyramids, where a base shape has a point above it. Other shapes can be built by stacking pyramids on top of prisms. In 1900, a special self-dual shape with hexagonal sides was discovered. Some self-dual shapes switch points and surfaces in interesting ways.

Main article: Hasse diagram
Main articles: regular form, canonical polyhedron, midsphere, elongated pyramids, prism, permutation, involution

Dual polytopes and tessellations

In geometry, the idea of a dual can be expanded to spaces with more dimensions, called dual polytopes. In two dimensions, these are known as dual polygons.

One polytope’s points match the higher-dimensional faces of another, and their lines connect in a matching way. For example, in four dimensions, the 600-cell has a shape called an icosahedron at each point; its dual, the 120-cell, has dodecahedra in the corresponding places, which are the dual shapes of icosahedra.