A 2D lattice gives an irregular honeycomb tessellation, with equal hexagons with point symmetry in the case of a regular triangular lattice it is regular in the case of a rectangular lattice the hexagons reduce to rectangles in rows and columns a square lattice gives the regular tessellation of squares note that the rectangles and the squares can also be generated by other lattices (for example the lattice defined by the vectors (1,0) and (1/2,1/2) gives squares).Voronoi tessellations of regular lattices of points in two or three dimensions give rise to many familiar tessellations. In general, a cross section of a 3D Voronoi tessellation is not a 2D Voronoi tessellation itself. Other equivalent names for this concept (or particular important cases of it): Voronoi polyhedra, Voronoi polygons, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), Dirichlet tessellation(s).Įxamples This is a slice of the Voronoi diagram of a random set of points in a 3D box. Voronoi diagrams that are used in geophysics and meteorology to analyse spatially distributed data (such as rainfall measurements) are called Thiessen polygons after American meteorologist Alfred H. Voronoi diagrams are named after Georgy Feodosievych Voronoy who defined and studied the general n-dimensional case in 1908. Peter Gustav Lejeune Dirichlet used two-dimensional and three-dimensional Voronoi diagrams in his study of quadratic forms in 1850.īritish physician John Snow used a Voronoi-like diagram in 1854 to illustrate how the majority of people who died in the Broad Street cholera outbreak lived closer to the infected Broad Street pump than to any other water pump. Informal use of Voronoi diagrams can be traced back to Descartes in 1644. As shown there, this property does not hold in general, even if the space is two-dimensional (but non-uniformly convex, and, in particular, non-Euclidean) and the sites are points. This is the geometric stability of Voronoi diagrams. Under relatively general conditions (the space is a possibly infinite-dimensional uniformly convex space, there can be infinitely many sites of a general form, etc.) Voronoi cells enjoy a certain stability property: a small change in the shapes of the sites, e.g., a change caused by some translation or distortion, yields a small change in the shape of the Voronoi cells.As shown there, this property does not necessarily hold when the distance is not attained. If the space is a normed space and the distance to each site is attained (e.g., when a site is a compact set or a closed ball), then each Voronoi cell can be represented as a union of line segments emanating from the sites.Then two points of the set are adjacent on the convex hull if and only if their Voronoi cells share an infinitely long side. Assume the setting is the Euclidean plane and a discrete set of points is given.The closest pair of points corresponds to two adjacent cells in the Voronoi diagram.The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites) corresponds to the Delaunay triangulation for the same set of points.The corresponding Voronoi diagrams look different for different distance metrics. In the simplest case, shown in the first picture, we are given a finite set of points. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. Voronoi cells are also known as Thiessen polygons. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation.

For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects.

Type of plane partition 20 points and their Voronoi cells (larger version below)

Nature tessellations