Dual Polyhedron - Self-dual Polyhedra

Self-dual Polyhedra

A self-dual polyhedron is a polyhedron whose dual is a congruent figure, though not necessarily the identical figure: for example, the dual of a regular tetrahedron is a regular tetrahedron "facing the opposite direction" (reflected through the origin).

A self-dual polyhedron must have the same number of vertices as faces. We can distinguish between structural (topological) duality and geometrical duality. The topological structure of a self-dual polyhedron is also self-dual. Whether or not such a polyhedron is also geometrically self-dual will depend on the particular geometrical duality being considered. For example, every polygon is topologically self-dual (it has the same number of vertices as edges, and these are switched by duality), but will not in general be geometrically self-dual (up to rigid motion, for instance) – regular polygons are geometrically self-dual (all angles are congruent, as are all edges, so under duality these congruences swap), but irregular polygons may not be geometrically self-dual.

The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity (average position) of the tangent points. If the polar reciprocal of the canonical form in the sphere is congruent to the original, then the figure is self-dual.

There are infinitely many self-dual polyhedra. The simplest infinite family are the pyramids of n sides and of canonical form. Another infinite family consists of polyhedra that can be roughly described as a pyramid sitting on top of a prism (with the same number of sides). Add a frustum (pyramid with the top cut off) below the prism and you get another infinite family, and so on.

There are many other convex, self-dual polyhedra. For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices.

Non-convex self-dual polyhedra can also be found, such as the excavated dodecahedron.