Research Interest

Self-Dual Maps

When the map of the borders of countries can also serve as the map of roads linking the capitals of those countries, that map is "self-dual".

Self-dual maps historically received little attention from the mathematical community, apart from an occasional reference that would offer a definition and a few examples. Perhaps simplicity of their description suggested that they weren't much more than graph-theoretical curiosities.

In 1988, I discovered a straightforward method by which one could recursively construct all "self-dual maps" on a surface, once a collection of primary examples were known. In 1989, for my undergraduate senior project, I examined all of a certain category of maps on the plane (ones that change from "border map" to "road map" without flipping), answering fundamental questions about the surprisingly orderly nature of such maps. (They come in only two flavors.) A couple of years later I refined and expanded my results to cover all (that is, even flippable) maps on the sphere, characterizing the types of maps by their symmetries (the various ways one could pick up the map and lay it back down upon itself) and their "dualities" (the various ways one could pick up the "border map" and lay it back down upon its "road map").

I derived a collection specialized results, and I intend to revisit them someday, but my enthusiasm for the subject waned after a colleague "scooped" me by publishing a paper on the subject.

Even so, I'm convinced there's a game or a Rubik's Cube-like puzzle in here somewhere.