can be constructed by connecting the face centers. In general this creates only a
topological dual.Images from
Kepler's
Harmonices Mundi (1619) 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 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 r and is centered at the origin (so that it is defined by the equation x^2 + y^2 + z^2 = r^2), then the polar dual of a convex polyhedron P is defined as {{Block indent|left=1.6|P^\circ = \{ q~\big|~q \cdot p \leq r^2 for all p in P \} ,}} where q \cdot p denotes the standard
dot product of q and p. Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning r=1 in the above definitions. For each face plane of P described by the linear equation x_0x + y_0y + z_0z = r^2, the corresponding vertex of the dual polyhedron P^\circ will have coordinates (x_0,y_0,z_0). Similarly, each vertex of P corresponds to a face plane of P^\circ, and each edge line of P corresponds to an edge line of P^\circ. The correspondence between the vertices, edges, and faces of P and P^\circ reverses inclusion. For example, if an edge of P contains a vertex, the corresponding edge of 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, 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, argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.
Canonical duals of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common
midsphere. 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. ==Self-dual polyhedra==