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Polyhedron associated with another by swapping vertices for faces

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. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original 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 polyhedrathe (convex) Platonic solids and (star) Kepler–Poinsot polyhedraform 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.

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

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.

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.

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.

Duality can be generalized to n dimensional space and dual polytopes; in two dimension these are called dual polygons. The vertices of one polytope correspond to the (n − 1) dimensional elements, or facets, of the other, and the j points that define a (j − 1) dimensional element will correspond to j hyperplanes that intersect to give a (n − j) dimensional element. The dual of an n dimensional tessellation or honeycomb can be defined similarly. In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the 600 cell is the icosahedron; the dual of the 600 cell is the 120 cell, whose facets are dodecahedra, which are the dual of the icosahedron.