# What is duality?

I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. Sometimes even applied to a method like simplex and dual simplex methods in linear programming.

My question is what is the general meaning behind the term duality and what is its relevance to mathematics. Do we mean the same always when we use the term? Or the examples I have put have no connection whatsoever?

Thanks a lot

Duality is a very general and broad concept, without a strict definition that captures all those uses. When applied to specific concepts, there usually is a precise definition for just that context. The common idea is that there are two things which basically are just two sides of the same coin.

Common themes in this topic include:

• Two different interpretations or descriptions of fundamentally the same structure or object
(e.g. roles of points and lines interchanged, roles of variables in LP changed)
• Primal and dual often are the same kind of object
(e.g. incidence configuration, vector space, linear program, planar graph, …)
• The dual of the dual is again the primal

Not every use of the word strictly satisfies all of these aspects, but the general idea usually is still the same.