An intrinsic property of a mathematical object which causes it to remain invariant under certain classes of transformations (such as rotation,
reflection, inversion,
or more abstract operations). The mathematical study of symmetry is systematized
and formalized in the extremely powerful and beautiful area of mathematics called
group theory.
Symmetry can be present in the form of coefficients of equations as well as in the physical arrangement of objects. By classifying the symmetry of polynomial equations
using the machinery of group theory, for example,
it is possible to prove the unsolvability of the general quintic
equation.
In physics, the extremely powerful Noether's symmetry theorem states that each symmetry of a system leads to a physically
conserved quantity. Symmetry under translation corresponds
to momentum conservation, symmetry under rotation to
angular momentum conservation, symmetry in time to energy conservation, etc.
