Dual polyhedron

Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. 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 dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an isotoxal polyhedron (one in which any two edges are equivalent [...]) is also isotoxal.

Duality is closely related to polar reciprocity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

Kinds of duality

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

Polar reciprocation

In Euclidean space, the dual of a polyhedron ${\displaystyle P}$ 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.

When the sphere has radius ${\displaystyle r}$ and is centered at the origin (so that it is defined by the equation ${\displaystyle x^2+y^2+z^2=r^2}$), then the polar dual of a convex polyhedron ${\displaystyle P}$ is defined as

${\displaystyle P^{\circ }=\{q|q\cdot p\le r^2}$ for all ${\displaystyle p}$ in ${\displaystyle P\},}$

where ${\displaystyle q\cdot p}$ denotes the standard dot product of ${\displaystyle q}$ and ${\displaystyle p}$.

Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning ${\displaystyle r=1}$ in the above definitions.

For each face plane of ${\displaystyle P}$ described by the linear equation $${\displaystyle x_{0}x+y_{0}y+z_{0}z=r^2,}$$ the corresponding vertex of the dual polyhedron ${\displaystyle P^{\circ }}$ will have coordinates ${\displaystyle (x_0,y_0,z_0)}$. Similarly, each vertex of ${\displaystyle P}$ corresponds to a face plane of ${\displaystyle P^{\circ }}$, and each edge line of ${\displaystyle P}$ corresponds to an edge line of ${\displaystyle P^{\circ }}$. The correspondence between the vertices, edges, and faces of ${\displaystyle P}$ and ${\displaystyle P^{\circ }}$ reverses inclusion. For example, if an edge of ${\displaystyle P}$ contains a vertex, the corresponding edge of ${\displaystyle P^{\circ }}$ will be contained in the corresponding face.

For a polyhedron with a center of symmetry, it is common to use a sphere centered on this point, as in the Dorman Luke construction (mentioned below). Failing that, for a polyhedron with a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents), this can be used. However, it is possible to reciprocate a polyhedron about any sphere, and the resulting form of the dual will depend on the size and position of the sphere; as the sphere is varied, so too is the dual form. 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).

The concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged. 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. 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.

Canonical duals

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. This form is unique up to congruences.

If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points, and thus will also be canonical. It is the canonical dual, and the two together form a canonical dual compound.

Dorman Luke construction

For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction.

Topological duality

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.

The vertices and edges of a convex polyhedron form a graph (the 1-skeleton of the polyhedron), embedded on the surface of the polyhedron (a topological sphere). This graph can be projected to form a Schlegel diagram on a flat plane. The graph formed by the vertices and edges of the dual polyhedron is the dual graph of the original graph.

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. Every such poset has a dual poset, formed by reversing all of the order relations. If the poset is visualized as a Hasse diagram, the dual poset can be visualized simply by turning the Hasse diagram upside down.

Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex geometric polyhedra, the dual polyhedra may not be realizable geometrically.