Voronoi Diagram - Examples

Examples

Voronoi tessellations of regular lattices of points in two or three dimensions give rise to many familiar tessellations.

• 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). See here for a dynamic visual example.
• A simple cubic lattice gives the cubic honeycomb.
• A hexagonal close-packed lattice gives a tesselation of space with trapezo-rhombic dodecahedra.
• A face-centred cubic lattice gives a tessellation of space with rhombic dodecahedra.
• A body-centred cubic lattice gives a tessellation of space with truncated octahedra.
• Parallel planes with regular triangular lattices aligned with each others' centers give the hexagonal prismatic honeycomb.
• Certain body centered tetragonal lattices give a tessellation of space with rhombo-hexagonal dodecahedra.

For the set of points (x, y) with x in a discrete set X and y in a discrete set Y, we get rectangular tiles with the points not necessarily at their centers.