Fundamental domain of a Euclidean lattice

A fundamental domain for a lattice $\Lambda\subset\mathbb R^n$ is a measurable set $F$ whose translates $F+\lambda$ , $\lambda\in\Lambda$ , partition $\mathbb R^n$ . A frequently used variant allows overlaps or omissions of measure zero; the convention must be specified.

A half-open domain

For $\Lambda=B\mathbb Z^n$ , the half-open parallelepiped $F=B[0,1)^n$ is an exact fundamental domain. Write each component of $B^{-1}x$ uniquely as an integer plus a number in $[0,1)$ . Its volume is $|\det B|$ , called the covolume of $\Lambda$ . Lattice translations give a concrete realization of the quotient $\mathbb R^n/\Lambda$ .