Inmathematics,continuous symmetry is an intuitive idea corresponding to the concept of viewing somesymmetries asmotions, as opposed todiscrete symmetry, e.g.reflection symmetry, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset of some higher-dimensional continuous symmetry, e.g. reflection of a 2-dimensional object in 3-dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.

The notion of continuous symmetry has largely and successfully been formalised in the mathematical notions oftopological group,Lie group andgroup action. For most practical purposes, continuous symmetry is modelled by agroup action of a topological group that preserves some structure. Particularly, let${\displaystyle f:X\to Y}$ be a function, and${\displaystyle G}$ is a group that acts on${\displaystyle X}$; then a subgroup${\displaystyle H\subseteq G}$ is a symmetry of${\displaystyle f}$ if${\displaystyle f(h\cdot x)=f(x)}$ for all${\displaystyle h\in H}$.

The simplest motions follow aone-parameter subgroup of a Lie group, such as theEuclidean group ofthree-dimensional space. For exampletranslation parallel to thex-axis byu units, asu varies, is a one-parameter group of motions. Rotation around thez-axis is also a one-parameter group.

Continuous symmetry has a basic role inNoether's theorem intheoretical physics, in the derivation ofconservation laws from symmetry principles, specifically for continuous symmetries. The search for continuous symmetries only intensified with the further developments ofquantum field theory.

• Goldstone's theorem
• Infinitesimal transformation
• Noether's theorem
• Sophus Lie
• Motion (geometry)
• Circular symmetry

• Barker, William H.; Howe, Roger (2007).Continuous Symmetry: from Euclid to Klein. American Mathematical Society.ISBN 978-0-8218-3900-3.

• Symmetry
• Lie groups
• Group actions

• Use American English from February 2019
• All Wikipedia articles written in American English
• Articles with short description
• Short description matches Wikidata