A planar graph $G$ is self-dual if it is isomorphic to its dual. We can study self-dual planar graphs by regarding the collection of all such isomorphisms as a coset in the automorphism group of a graph which may be constructed from $G$ and $G^{*}$. For many infinite self-dual graphs, this automorphism group is related to the 17 wallpaper groups. We will use these tools to describe and construct self-dual graphs and tilings which are Euclidean in the sense that their automorphism group has a free abelian subgroup of rank two with finite index.
An application of this extended concept of symmetry is the creation of artwork with a balance between color and form, a work, for example, in which a photographic negative would be indistinguishable from the actual picture of the work itself.