What does the Voronoi Diagram (also called Dirichlet tessellation) look like?
A Voronoi diagram is a geometric partitioning of a plane into regions based on distance to a specified set of discrete points, called seeds or sites. Each region, known as a Voronoi cell, consists of all points in the plane that are closer to its associated seed than to any other. The resulting structure is a network of convex polygons that tile the space, meeting at edges and vertices. The edges of these polygons are the perpendicular bisectors of the line segments connecting neighboring seeds, and the vertices are points equidistant to three or more seeds. In its most fundamental form for a set of points in a Euclidean plane, the diagram resembles a honeycomb-like mosaic of irregular polygonal cells, where each cell's shape and size are determined entirely by the proximity and spatial arrangement of the surrounding seeds.
The visual characteristics of the diagram are governed by the distribution of the seed points. If the seeds are randomly scattered, the polygons will be irregular and varied in size, creating a seemingly chaotic but mathematically precise patchwork. In contrast, if the seeds are arranged in a perfectly regular lattice pattern—such as a square or hexagonal grid—the Voronoi cells will themselves become uniform squares or regular hexagons, respectively. The boundaries between cells are straight line segments in classical Euclidean space, but the concept extends to other metrics, where edges may become curved; for instance, using a Manhattan distance metric produces boundaries composed of angled line segments. In three dimensions, the diagram comprises convex polyhedra that fill space, analogous to the polygonal cells in two dimensions.
The diagram's appearance is not merely an abstract geometric construct; it is a direct visual manifestation of proximity and influence. Each cell's boundaries delineate the region of dominance for its seed, making the structure a natural model for territories in spatial competition. This is why Voronoi diagrams visually resemble patterns observed in nature and human systems, such as the cracking of dried mud, the structure of giraffe skin patches, the zones of control around cellular towers in a network coverage map, or the market areas around competing retail stores. The intersections where three or more edges meet—the vertices—are critical points of equilibrium, being maximally distant from the nearest seeds, and their arrangement dictates the overall connectivity of the cellular network.
Understanding the visual outcome requires recognizing that the diagram is the dual graph of the Delaunay triangulation, where seeds are connected if their Voronoi cells share an edge. This duality means that the triangulation's edges are perpendicular to the Voronoi edges, providing a computational framework for its construction and analysis. The immediate implication of its structure is that any change in the seed set—adding, removing, or moving a single point—causes a local but potentially cascading reorganization of the cell boundaries, altering the tessellation's topology. Thus, the Voronoi diagram's appearance is a dynamic, data-driven map of nearest-neighbor relationships, whose geometry encodes both the discrete locations of the seeds and the continuous metric space they inhabit.