Translational symmetry

if the result after applying A doesn't change if the argument function is translated.

Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space.

According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law.

Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set {p + na | n ∈ Z} = p + Z a.

For each set of k independent translation vectors, the symmetry group is isomorphic with Zk.

Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1.

This parallelepiped is a fundamental region of the symmetry: any pattern on or in it is possible, and this defines the whole object.

In general in 2D, we can take pa + qb and ra + sb for integers p, q, r, and s such that ps − qr is 1 or −1.

Each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product.