A unified approach to the topological origin of Wigner–Seitz cells

While Wigner-Seitz cells of different Bravais lattices exhibit distinct geometric features, their topological formation follows a unified fundamental principle. This study explores the topological origins of Wigner-Seitz cells. We demonstrate that a finite set of lattice vectors, termed root vectors, can universally and fully characterize these cells in n-dimensional Euclidean space (\mathbbR^n) via half-space representations. The configuration of root vectors determines the topological shapes of the corresponding Wigner-Seitz cells. Specifically, the number of orthogonal pairs within the base vectors dictates the degeneracy of codimension-1 faces. For general lattices with no orthogonal pairs, the Wigner-Seitz cell possesses the maximum possible number of codimension-1 faces.