Seven Bridges of Königsberg
Recognizable planar dual graphs, outside the context of polyhedra, appeared as early as 1725, in Pierre Varignon's posthumously published work, "Nouvelle Méchanique ou Statique". This was even before Leonhard Euler's 1736 work on the Seven Bridges of Königsberg that is often taken to be the first work on graph theory. Varignon analyzed the forces on static systems of struts by drawing a graph dual to the struts, with edge lengths proportional to the forces on the struts; this dual graph is a type of Cremona diagram. In connection with the four color theorem, the dual graphs of maps (subdivisions of the plane into regions) were mentioned by Alfred Kempe in 1879, and extended to maps on non-planar surfaces by in 1891. Duality as an operation on abstract planar graphs was introduced by Hassler Whitney in 1931.