Translational Symmetry

In geometry, a translation "slides" an object by a a: Ta(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 invariance under discrete translation.

Analogously an operator A on functions is said to be translation 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 if they do not distinguish different points in space. According to Noether's theorem, 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