Brillouin zones

A region in an ordinary Voronoi diagram shows where a given point is closest. One can also consider higher-order Voronoi diagrams in which each region shows where a given point is the k^^{th} closest. The total area of each region is the same for every k, but some complexity in shape is seen, though for large k they always in a sense approximate circles. 3D versions of such regions have been encountered in studies of quantum mechanical properties of crystals since the 1930s.