Rotational symmetry

Rotational symmetry is a term meaning the symmetry of an object relative to all or some of its own rotations of an m- dimensional Euclidean space . Own rotations are called isometric varieties that preserve orientation. Thus, the symmetry group corresponding to rotations is a subgroup of the group E ⁺ ( m ) (see Euclidean group ).

Translational symmetry can be considered as a special case of rotational symmetry - rotation around an infinitely distant point. With this generalization, the group of rotational symmetry coincides with the full E ⁺ ( m ). This kind of symmetry is not applicable to finite objects, since it makes the whole space homogeneous, but it is used in the formulation of physical laws.

The set of proper rotations around a fixed point in space form a special orthogonal group SO (m) - a group of m × m orthogonal matrices with a determinant of 1. For a special case m = 3, the group has a special name - a group of rotations .

In physics, invariance with respect to a group of rotations is called isotropy of space (all directions in space are equal) and is expressed in the invariance of physical laws, in particular, equations of motion, with respect to rotations. Noether's theorem connects this invariance with the presence of a conserved quantity (integral of motion) - the angular momentum .