Editing Dual polyhedron (section)

{{Short description|Polyhedron associated with another by swapping vertices for faces}}
[[File:Polyhedron pair 6-8.png|thumb|right|upright=1.35|The dual of a [[cube]] is an [[octahedron]]. Vertices of one correspond to faces of the other, and edges correspond to each other.]]

In [[geometry]], every [[polyhedron]] is associated with a second '''dual''' structure, where the [[Vertex (geometry)|vertices]] of one correspond to the [[Face (geometry)|faces]] of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.<ref>{{harvtxt|Wenninger|1983}}, "Basic notions about stellation and duality", p.&nbsp;1.</ref> Such dual figures remain combinatorial or [[Abstract polytope|abstract polyhedra]], but not all can also be constructed as geometric polyhedra.<ref>{{harvtxt|Grünbaum|2003}}</ref> Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Duality preserves the [[Symmetry|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{{snd}}the (convex) [[Platonic solid]]s and (star) [[Kepler–Poinsot polyhedra]]{{snd}}form dual pairs, where the regular [[tetrahedron]] is [[#Self-dual polyhedra|self-dual]]. The dual of an [[Isogonal figure|isogonal]] polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an [[Isohedral figure|isohedral]] polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an [[Isotoxal figure|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.