Chapter 7: Mechanisms in Programs and Nature

Section 8: The Problem of Satisfying Constraints

Voronoi diagrams

The Voronoi diagram for a set of points shows the region around each point in which one is closer to that point than to any other. (The edges of the regions are thus like watersheds.) The pictures below show a few examples. In 2D the regions in a Voronoi diagram are always polygons, and in 3D polyhedra. If all the points lie on a repetitive lattice each region will always be the same, and is often known as a Wigner-Seitz cell or a Dirichlet domain. For a simple cubic lattice the regions are cubes with 6 faces. For an fcc lattice they are rhombic dodecahedra with 12 faces and for a bcc lattice they are truncated octahedra (tetradecahedra) with 14 faces. (Compare page 929.)

Voronoi diagrams for irregularly distributed points have found many applications. In 2D they are used in studies of animal territories, retail store utilization and municipal districting. In 3D they are used as simple models of foams, grains in solids, assemblies of biological cells and self-gravitating regions in primordial galaxy formation. Voronoi diagrams are relevant whenever there is growth in all directions at an identical speed from a collection of seed points. (In high dimensions they also appear immediately in studying error-correcting codes.)

Modern computational geometry has provided efficient algorithms for constructing Voronoi diagrams, and has allowed them to be used in mesh generation, point location, cluster analysis, machining plans and many other computational tasks.

From Stephen Wolfram: A New Kind of Science [citation]