symmetries

Symmetry is a property of an object or system that remains unchanged under a set of transformations. When a transformation maps the object onto itself or preserves a key attribute, the object is said to have symmetry. Symmetries can be geometric, physical, or abstract, and they are central to both mathematics and the sciences because they reveal invariants and simplify complex structures.

Geometric symmetries include translations, rotations, reflections, and glide reflections. A circle has continuous rotational symmetry, a square has discrete rotational and reflectional symmetries, and a lattice may exhibit translational symmetry. In higher dimensions, symmetries can be described by groups of transformations; the set of all symmetries of an object forms a symmetry group.

In physics, symmetries are statements about invariance of laws under transformations. For example, shifting the origin in time or space leaves the laws unchanged, leading to conservation laws via Noether's theorem. Translational symmetry implies momentum conservation; rotational symmetry implies angular momentum conservation; time-translation symmetry implies energy conservation. Gauge symmetries underpin the standard model of particle physics and play a central role in field theories.

Symmetry has practical consequences across disciplines. In chemistry, molecular shapes and spectra reflect symmetries; in crystallography, space groups classify crystal patterns; in art and music, symmetry informs composition and structure.

There are global symmetries, acting on all parts of a system, and local or gauge symmetries, which may vary from point to point in space-time. With group theory, symmetries are studied as mathematical objects, offering a unifying language for patterns and invariants.