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Dual polyhedron

Polyhedron associated with another by swapping vertices for faces / From Wikipedia, the free encyclopedia

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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.[1] Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra.[2] Starting with any given polyhedron, the dual of its dual is the original polyhedron.

The dual of a cube is an octahedron. Vertices of one correspond to faces of the other, and edges correspond to each other.

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.