A graph on the plane is a set of points linked by edges. If two points are linked we call them neighbors. Harmonic graphs are those that have the property that each point is exactly in the center (barycenter) of its neighbors. Harmonic graphs usually look “harmonic”, despite the name has nothing to do with this fact.
The pictures in this gallery are the outcome of research on harmonic random graphs. Given a random set of points in the plane we can construct a graph linking the points that are “close” to each other. Is it possible to deform this graph to turn it into a harmonic one without moving too much the points? The answer to this question has important consequences in probability theory and other fields. More info.
Harmonic Voronoi Tessellation
This is the Voronoi tessellation of a random harmonic graph. The Voronoi tessellation is a way to divide the plane in regions. Given a set of “sites” in the plane, we assign to each site the region composed of all the points in the plane that are closer to this site than to the others. The lines in the picture corresponds to party walls.
Harmonic Delaunay Triangulation
This is a random harmonic graph constructed from a Delaunay triangulation of a random set of points in the plane. The Delaunay triangulation is a way to decide which points are neighbors and which are not: three points are neighbors if the circle that they determine do not contain any other point.
Islands or Sea?
The Harmonic Delaunay Triangulation (previous picture) was constructed in such a way that each point is at the center of its neighbors. To do that, each point was moved a little bit from its original location. This picture represents with the blue color those points that were moved to the right and with red those that were moved to the left.
Are the red regions islands in a blue sea? Or maybe they are blue islands in a red sea.
There is a theorem that states that none of them happens: all of them are islands and there is no sea. This means that there is no way to go from a given point to infinity using only one color.