Dual polyhedron | EPFL Graph Search

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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. 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 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. 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 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 and is centered at the origin (so that it is defined by the equation ), then the polar dual of a convex polyhedron is defined as where denotes the standard dot product of and .

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