Symmetry (geometry)
Symmetry is invariance under geometric transformations.
In geometry, an object has symmetry if there is an operation or transformation—such as translation, scaling, rotation, or reflection—that maps the figure onto itself, meaning the object is invariant under the transform. Symmetry can be thought of as an immunity to change, and the set of transforms under which an object is symmetric forms a mathematical group called the symmetry group of the object.
- field
- Geometry
- known_for
- Symmetry as invariance under transformations, including reflectional, rotational, translational, glide reflection, and rotoreflection symmetries
Lore & Background
The most common group of transforms applied to objects is the Euclidean group of isometries, which are distance-preserving transformations in space, including reflections, rotations, translations, and combinations. Under an isometric transformation, a geometric object is symmetric if after transformation it is indistinguishable from the original. A geometric object is typically symmetric only under a subgroup of all isometries. By the Cartan–Dieudonné theorem, an orthogonal transformation in n-dimensional space can be represented by the composition of at most n reflections.
Reader's Guide
Reflectional symmetry, also called mirror symmetry or bilateral symmetry, involves reflection across a line in two dimensions or a plane in three dimensions. A square has four axes of symmetry, while a circle has infinitely many. Rotational symmetry involves rotations in m-dimensional Euclidean space; rotations are direct isometries that preserve orientation. Translational symmetry leaves an object invariant under discrete or continuous translations. Glide reflection symmetry combines a reflection with a translation along the line or plane. Rotoreflection symmetry in 3D is a rotation about an axis combined with reflection in a perpendicular plane. The symmetry groups associated with these transforms are fundamental to classifying geometric objects and understanding physical laws, such as the conservation of angular momentum via Noether's theorem.
Did You Know?
- A circle rotated about its center has rotational symmetry because all points before and after the transform are indistinguishable.
- The triangles with reflection symmetry are isosceles; the quadrilaterals with this symmetry are kites and isosceles trapezoids.
- A point reflection in three dimensions changes a left-handed coordinate system into a right-handed one; note that 'left-right symmetry' typically refers to mirror symmetry across a plane, not point reflection.
- The composition of two glide reflections results in a translation symmetry with twice the translation vector.
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