text: The concept of Voronoi decomposition was investigated by Peter Gustav Lejeune Dirichlet, leading to the name Dirichlet domain. Further contributions were made from Evgraf Fedorov, (Fedorov parallelohedron), Georgy Voronoy (Voronoi polyhedron), and Paul Niggli (Wirkungsbereich).
  The application to condensed matter physics was first proposed by Eugene Wigner and Frederick Seitz in a 1933 paper, where it was used to solve the Schrödinger equation for free electrons in elemental sodium. They approximated the shape of the Wigner–Seitz cell in sodium, which is a truncated octahedron, as a sphere of equal volume, and solved the Schrödinger equation exactly using periodic boundary conditions, which require  at the surface of the sphere. A similar calculation which also accounted for the non-spherical nature of the Wigner–Seitz cell was performed later by John C. Slater.
  There are only five topologically distinct polyhedra which tile three-dimensional space, . These are referred to as the parallelohedra. They are the subject of mathematical interest, such as in higher dimensions. These five parallelohedra can be used to classify the three dimensional lattices using the concept of a projective plane, as suggested by John Horton Conway and Neil Sloane. However, while a topological classification considers any affine transformation to lead to an identical class, a more specific classification leads to 24 distinct classes of voronoi polyhedra with parallel edges which tile space. For example, the rectangular cuboid, right square prism, and cube belong to the same topological class, but are distinguished by different ratios of their sides. This classification of the  24 types of voronoi polyhedra for Bravais lattices was first laid out by Boris Delaunay.