Duality (mathematics)

In many areas of mathematics, there is the following situation: for each object of each class under consideration, there is a dual object whose dual object is in turn or at least very close. Frequently, there is also a connection between and, which describes the relationship between them closer.

Duality in geometry

Duality of polytopes

Two polytopes (ie, polygons, polyhedra, etc.) and hot abstract dual if their side groups ( the inclusion of its sides, corners, edges, faces, etc.) are antiisomorph. For example: If you choose the midpoints of the faces of a three-dimensional convex polyhedron as corners, and you just connect two "new" vertices if the two corresponding side faces of a common edge have, you get a dual polyhedron. The number of vertices of is equal to the area of number and vice versa, the edges numbers are the same.

But this says nothing about whether the polytopes and are invariant under the same symmetry pictures. A square and a rectangle are any example, an abstract dual, because at every turn meet two edges and each edge has two vertices. In general, the symmetry of the quadrilateral figures already include no reflections, for the square on the other hand.

There is a special combinatorial dual polytope, called the Polar For each polytope. For this one summarizes the polytope as a closed subset of a Euclidean vector space. Polars then consists of all points that satisfy all of the inequality. With the proviso that the geometrical center of gravity of for the zero, and have the same symmetry of its polar group. The double -dual polyhedron is too similar and equal to this, when the zero point is contained in its interior.

For examples, see: Platonic solid, Archimedean body

Duality principle of projective geometry and incidence structures

In projective geometry the following duality principle: you Swaps in a true statement about points and lines of a projective plane, the terms " point " and " straight " and substituting in each case the term " straight line connecting two points " with the term " intersection of two lines " and vice versa, we obtain again a true statement about the dual projective geometry. For desarguesian projective geometries, ie for example, all two-dimensional projective spaces over fields the dual projective geometry is up to isomorphism identical to the original geometry, ie the dual rate in these geometries here is equivalent to the original set.

Examples of pairs of dual sets are the set of Desargues and its reversal or set of Pascal and the set of Brianchon.

→ The concrete construction of the duality isomorphism as on a projective space depends on the selected projective coordinate system and is thus presented projective coordinate system in the main article.

→ A generalization of the duality principle in projective geometry is the duality principle for incidence structures.

Geometric dual graph

A similar definition also knows the graph theory for planar graphs. A dual graph to the graph geometry is created by new nodes are added to each face of the graph, and for each edge a new edge is created which connects the two adjacent surfaces of the.

The graph is not only planar but also connected, as is also true here that the number of nodes in the number of areas corresponds to the number of surfaces to that of the node and the number of edges is constant. In the coherent case, there is thus bijective mappings between the edge sets of the two graphs and the amounts of each node and surfaces. In addition, true that.

Dual space of a vector space

Is a vector space over a field, then the dual vector space or dual space of the vector space whose elements are the linear maps. Is finite, so has the same dimension as, and is canonically isomorphic to.

In the case of a Banach space is the dual space of the continuous linear functionals. Is infinite dimensional, the Bidualraum is generally not canonically isomorphic to, however, there is a canonical embedding in Bidualraum. Those areas for which this embedding is surjective (and hence an isomorphism ) are called reflexive. Examples are the spaces Lp for and all Hilbert spaces.

Set theory: Complement

A duality, which is not normally referred to with this word, is the formation of the complement of a lot: Is a basic amount given, is the complement of a subset of the set of elements of which are not in. The complement of the complement is again itself The complementation is set union and intersection related to each other: (see de Morgan's rules).

A generalization of this example, the negation in any Boolean algebra dar.

After the duality principle for associations obtained from any true statement about subsets of a ground set back a true statement, if one exchanges the symbols ( union ) and (intersection ) and the symbols ( empty set) and (basic amount ).

• See also: Complement ( set theory ), Boolean algebra