Rotation group

A group of rotations that preserves a geometric object or space. Its elements are rotations, and its operation is composition.

Group (mathematics): A set equipped with an associative operation, an identity element, and inverses for every element. A rotation group is a group whose elements are geometric rotations.
Composition of rotations: The operation of applying one rotation after another, producing a single resulting rotation. Composition is the group operation that combines rotations.
Cyclic group: A group generated by repeated application of one element. Rotations of a regular polygon about its center form a finite cyclic group.
SO(2): The group of all rotations of the Euclidean plane about the origin. It models continuous planar rotational symmetry.
Reflection (mathematics): A transformation that reverses orientation by mirroring points across a line, plane, or hyperplane. Reflections preserve distances but are not proper rotations.
Rotation (mathematics): A transformation that turns points around a fixed center, axis, or subspace while preserving distances. Rotations supply the transformations that form the group.
Rotation matrix: An orthogonal matrix with determinant one that represents a proper rotation in Euclidean space. Matrices provide a concrete way to calculate and compose rotations.
Dihedral group: The symmetry group of a regular polygon, including rotations and reflections. Its full symmetry group adds reflections to the polygon's rotation subgroup.
SO(3) representation theory: The study of linear actions of the three-dimensional rotation group on vector spaces. Its representations classify how physical quantities transform under spatial rotations.
Euclidean group: The group of all distance-preserving transformations of Euclidean space, including rotations and translations. It broadens rotational symmetry to include translations and other isometries.

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