# Geometric duality

## Geometric duality Topics

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### Duality principle

All the propositions in projective geometry occur in dual pairs which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line." The principle was enunciated by Gergonne (1825-1826; Cremona 1960, p. x). A similar duality exists for reciprocation as first enunciated by Poncelet (1817-1818; Casey 1893; Lachlan 1893; Cremona 1960, p. x).Examples of dual geometric objects include Brianchon's theorem and Pascal's theorem, the 15 Plücker lines and 15 Salmon points, the 20 Cayley lines and 20 Steiner points, the 60 Pascal lines and 60 Kirkman points, dual polyhedra, and dual tessellations.Propositions which are equivalent to their duals are said to be self-dual...

### Dual polyhedron

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..