Fundamental domain

Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral.

Frequently D is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero.

For example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus.

For example, for wallpaper groups the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell.

Thus, the free regular set in this example is The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom including the point in the middle: The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author.

A lattice in the complex plane and its fundamental domain, with quotient a torus.
Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.