The red object is an example of an exceptionally symmetric fundamental domain. It is a fundamental domain of the square lattice, in the sense that every point of the plane has a representative within the red region, where point and representative are related by a unique lattice translation. In geometrical terms, one says the fundamental domain tiles the plane, that is, translates (by the square lattice) cover the plane without overlaps. Somewhat more to the point is the topological statement that, by means of identifications provided by two generating (primitive) translations of the square lattice, the boundary of the red region can be sewn together to form a torus.

The geometrical shape of a fudamental domain is far from unique; this particular shape, however, is exceptionally symmetric in the sense that it has more symmetries than its parent lattice. One of the geometrically simplest fundamental domains is the Voronoi domain, defined to be the set of points of the plane for which the origin is the closest lattice point. By construction, Voronoi domains have the same symmetry as the lattice. The Voronoi domain of the square lattice is itself a square and has four-fold rotational symmetry. It's clear from the image that the red region is eight-fold symmetric. This implies that there are actually two distinct ways of tiling the plane with this tile, related by a 45° rotation (or equivalently, two ways of sewing the boundary to make a torus).

The analog for the triangular lattice, a twelve-fold symmetric fundamental region, also exists and was discovered in the context of quasiperiodic tilings by my former student Eric Cockayne.