Dimension-reversing Dualities
There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an i-dimensional feature of an n-dimensional polytope corresponding to an (n − i − 1)-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure.
From any three-dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph.
In topology, Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological manifold is respresented as a cell complex, then the dual of the complex (a higher dimensional generalization of the planar graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the kth homology group and the (n − k)th cohomology group.
Another example of a dimension-reversing duality arises in projective geometry. In the projective plane, it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to a point, in an incidence-preserving way: in terms of the incidence matrix of the points and lines in the plane, this operation is just that of forming the transpose. Transformations of this type exist also in any higher dimension; one way to construct them is to use the same polar transformations that generate polyhedron and polytope duality. Due to this ability to replace any configuration of points and lines with a corresponding configuration of lines and points, there arises a general principle of duality in projective geometry: given any theorem in plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. A simple example is that the statement “two points determine a unique line, the line passing through these points” has the dual statement that “two lines determine a unique point, the intersection point of these two lines”. For further examples, see Dual theorems.
The points, lines, and higher dimensional subspaces n-dimensional projective space may be interpreted as describing the linear subspaces of an (n + 1)-dimensional vector space; if this vector space is supplied with an inner product the transformation from any linear subspace to its perpendicular subspace is an example of a projective duality. The Hodge dual extends this duality within an inner product space by providing a canonical correspondence between the elements of the exterior algebra.
A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space Rn), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.
Read more about this topic: Duality (mathematics)