By the duality principle, for every polyhedron, there exists another polyhedron in which faces and polyhedron vertices occupy complementary locations. This polyhedron is known as the dual, or reciprocal. The process of taking the dual is also called reciprocation, or polar reciprocation. Brückner (1900) was among the first to give a precise definition of duality (Wenninger 1983, p. 1).
Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Any polyhedron can be associated with a second (abstract, combinatorial, topological) dual figure, 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. Even when a pair of polyhedra cannot be obtained by reciprocation, they may be called (abstract, combinatorial, or topogical) 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. However, not all such duals are geometric polyhedra.
The dual operation is implemented in the Wolfram Language as DualPolyhedron[poly].
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The dual of a Platonic solid, Archimedean solid, or in fact any uniform polyhedron can be computed by connecting the midpoints of the sides surrounding each polyhedron vertex (the vertex figure; left figure), and constructing the corresponding tangential polygon (tangent to the circumcircle of the vertex figure; right figure). This is sometimes called the Dorman-Luke construction (Wenninger 1983, p. 30; Cundy and Rollett 1989, p. 117). Dorman Luke's construction can only be used where a polyhedron has such a midsphere and the vertex figure is cyclic.
According to Cundy and Rollett (1989, p. 79), in reciprocation, polar lines are perpendicular, and, for a suitable choice of radius, they can be made to intersect. This is the most interesting position in which reciprocal polyhedra can be placed, with each edge of one intersecting at right angles (and usually also at the mid-point) the corresponding edge of the other. In fact, many attractive polyhedron compounds are formed in precisely this way.
The dual polyhedron of a Platonic solid or Archimedean solid can be also drawn by constructing polyhedron
edges tangent to the midsphere (sometimes also
known as the reciprocating sphere or intersphere) which are perpendicular
to the original polyhedron edges. Furthermore,
let 
be the inradius of the dual polyhedron (corresponding
to the insphere, which touches the faces of the dual
solid), 
be the midradius of both the polyhedron and its dual
(corresponding to the midsphere, which touches the
edges of both the polyhedron and its duals), and 
the circumradius (corresponding
to the circumsphere of the solid which touches the
vertices of the solid). Since the circumsphere and
insphere are dual to each other, 
, 
,
and 
obey the polar relationship
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(Cundy and Rollett 1989, Table II following p. 144).
The process of forming duals via reciprocating with respect to the midsphere is illustrated above for the Platonic solids. The top row shows the original solids. The middle row shows the vertex figures of the original solid as lines superposed on the tangential polygons forming the corresponding duals. Finally, the polyhedron compounds consisting of a polyhedron and its dual are illustrated in the bottom row.
For an Archimedean solid with 
vertices, 
faces, and 
edges, the dual polyhedron has 
vertices, 
faces, and 
edges. The dual of an isogonal solid (i.e., all vertices are
alike) is isohedral (i.e., all faces are alike) (Wenninger 1983, p. 5).
The dual of any non-convex uniform polyhedron is a stellated form of the convex hull of the given polyhedron (Wenninger 1983, pp. 3-4 and 40).
For a Platonic or Archimedean solid, the ratio of the volume of the solid and its dual is the same as the ratio of the surface area of the solid and its dual, a property first noted by Apollonius for the dodecahedron and icosahedron.
The following table gives a list of the duals of the Platonic solids and Kepler-Poinsot polyhedra, together with the names of the polyhedron-dual compounds. (Note that the duals of the Platonic solids are themselves Platonic solids, so no new solids are formed by taking the duals of the Platonic solids.)
Duals can also be taken of other polyhedrons, including the Archimedean solids and uniform solids. The names of some solids and their duals are given in the table below.
When a polychoron with Schläfli symbol 
and its dual are in reciprocal positions, the vertices of 
's bounding polyhedra can be found by selecting those
vertices of 
closest to each vertex of 
.
