the Wigner-Seitz cell
/ Wigner-Seitz -> VIG-ner ZYTES /
There is a way to draw a primitive cell that is both smallest AND as symmetric as the lattice itself, with no arbitrary choice of axes. Pick one lattice point and claim all the space that is closer to it than to any other lattice point. That territory is the Wigner-Seitz cell — the home region of a lattice point, like the catchment area of the nearest school.
Construction recipe: draw lines from your chosen point to all its neighbours, then draw the perpendicular-bisector plane of each line; the smallest volume enclosed by these planes is the cell. It contains exactly one lattice point (so it is primitive) and it inherits the full point symmetry of the lattice. For example, the Wigner-Seitz cell of BCC is a truncated octahedron; that of FCC is a rhombic dodecahedron.
The Wigner-Seitz cell is the real-space twin of the Brillouin zone — do the identical bisector construction on the reciprocal lattice and you get the first Brillouin zone, the central object of electronic band theory. It is also called the Voronoi (or Dirichlet) cell in mathematics; the same idea partitions any set of points into nearest-neighbour regions.
For a 1D chain of points spaced a apart, the Wigner-Seitz cell is just the segment from -a/2 to +a/2 around a point — everything closer to that point than to its neighbours. In 2D a square lattice gives a square WS cell; a triangular lattice gives a regular hexagon.
The WS cell is the region of space nearest to one lattice point.
Applied in real space the construction gives a symmetric primitive cell; applied identically to the reciprocal lattice it gives the first Brillouin zone. The recipe is the same; only which lattice you feed it changes.