In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the mathematical notions of topological group, Lie group and group action. For most practical purposes continuous symmetry is modelled by a group action of a topological group. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In physics, motion means a change in the position of a body with respect to time, as measured by a particular observer in a particular frame of reference. ... Figures with the axes of symmetry drawn in. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a symmetry group describes all symmetries of objects. ...

The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space. For example translation parallel to the x-axis by u units, as u varies, is a one-parameter group of motions. Rotation around the z-axis is also a one-parameter group. In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is injective... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... The space we live in is three-dimensional space. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...

Continuous symmetry has a basis role in Noether's theorem in theoretical physics, in the derivation of conservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments of quantum field theory. Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between the symmetries and the conservation laws. ... Theoretical physics is physics that employs mathematical models and abstractions rather than experimental processes. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

See also

• Infinitesimal transformation
• Sophus Lie