Бұрылу симметриясы

Property of objects which appear unchanged after a partial rotation

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.

Formally the rotational symmetry is symmetry with respect to some or all rotations in m dimensional Euclidean space. Rotations are direct isometries, i. e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E^( +)(m) (see Euclidean group). Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole E(m). With the modified notion of symmetry for vector fields the symmetry group can also be E^( +)(m). For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m × m orthogonal matrices with determinant 1. For 1=m = 3 this is the rotation group SO(3). In another definition of the word, the rotation group of an object is the symmetry group within E^( +)(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group. Laws of physics are SO(3) invariant if they do not distinguish different directions in space. Because of Noether's theorem, the rotational symmetry of a physical system is equivalent to the angular momentum conservation law.

Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The fundamental domain is a half line. In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half plane through the axis, and a radial half line, respectively. Axisymmetric and axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i. e. rotational symmetry with respect to a central axis) like a doughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties). In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e. g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e. g. the duocylinder and various regular duoprisms.