# Translational symmetry

In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by **a**: *T*_{a}(**p**) = **p** + **a**.

In physics and mathematics, continuous **translational symmetry** is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation.

Analogously an operator *A* on functions is said to be translationally invariant with respect to a translation operator if the result after applying *A* doesn't change if the argument function is translated.
More precisely it must hold that

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 symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.