Dual polyhedron

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.

For example, the regular polyhedra – the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual.

The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.

is often defined in terms of polar reciprocation about a sphere.

Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius.

The choice of center for the sphere is sufficient to define the dual up to similarity.

If a polyhedron in Euclidean space has a face plane, edge line, or vertex lying on the center of the sphere, the corresponding element of its dual will go to infinity.

Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required 'plane at infinity'.

Some theorists prefer to stick to Euclidean space and say that there is no dual.

Meanwhile, Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion).

Projective polarity works well enough for convex polyhedra.

But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear.

[5] Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum (2007) argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.

Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere (or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere.

[7] Even when a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way.

Such pairs of polyhedra are still topologically or abstractly dual.

This graph can be projected to form a Schlegel diagram on a flat plane.

More generally, for any polyhedron whose faces form a closed surface, the vertices and edges of the polyhedron form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph of the original graph.

An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that incidences, or connections, between elements of the set correspond to incidences between elements (faces, edges, vertices) of a polyhedron.

If the poset is visualized as a Hasse diagram, the dual poset can be visualized simply by turning the Hasse diagram upside down.

Topologically, a polyhedron is said to be self-dual if its dual has exactly the same connectivity between vertices, edges, and faces.

Geometrically, it is not only topologically self-dual, but its polar reciprocal about a certain point, typically its centroid, is a similar figure.

Every polygon is topologically self-dual, since it has the same number of vertices as edges, and these are switched by duality.

But it is not necessarily self-dual (up to rigid motion, for instance).

Every polygon has a regular form which is geometrically self-dual about its intersphere: all angles are congruent, as are all edges, so under duality these congruences swap.

[8] Another infinite family, elongated pyramids, consists of polyhedra that can be roughly described as a pyramid sitting on top of a prism (with the same number of sides).

Adding a frustum (pyramid with the top cut off) below the prism generates another infinite family, and so on.

[9] A self-dual non-convex icosahedron with hexagonal faces was identified by Brückner in 1900.

The dual of an n-dimensional tessellation or honeycomb can be defined similarly.

For the polar reciprocals of the regular and uniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure.