Abstractly, a spatial configuration is said to possess rotational symmetry if remains invariant under the group . Here, denotes the group of rotations of and is viewed as a subgroup of the automorphism group of all automorphisms which leave unchanged. A more intuitive definition of rotational symmetry comes from the case of planar figures in Cartesian space.

For arbitrary, a geometric object in is said to possess rotational symmetry if there exists a point so that the object, when rotated a certain number of degrees (or radians) about said point, looks precisely the same as it did originally. This notion can be made more precise by counting the number of distinct ways the object can be rotated to look like itself; this number is called the degree or the order of the symmetry.

Rotational symmetry of degree corresponds to a plane figure being the same when rotated by degrees, or by radians.

The regular pentagon in the figure above has a rotational symmetry of order 5 due to the fact that rotating it about the center point by radians, , yields the exact same figure. This is a particular example of a more general fact, namely that any regular planar n-gon has rotational symmetry of order . In the case of the regular planar -gon, the collection of all such symmetries is a group denoted by , is isomorphic to the cyclic group of integers modulo , and is a proper subgroup of the dihedral group of all symmetries--rotational and otherwise--of the figure.

By the definition given above, rotational symmetry of degree 1 corresponds to an object having symmetry about a point only when rotated by degrees; clearly, this condition is satisfied only by objects which have no symmetry, i.e., those objects whose rotational symmetry group is trivial. Therefore, the simplest possible rotational symmetry is of order 2 and is possessed, e.g., by planar parallelograms.

In some literature, rotational symmetry of order is defined by classifying the results of rotating a figure about a line rather than about a point (Weyl 1982). In particular, such sources define a figure to have rotational symmetry of order if the figure which remains identical after a -radian rotation about (which is called the axis of rotation). These two perspectives yield the same result, however; for example, in the figure above, the -radian clockwise rotation of the pentagon about its center point can equivalently be viewed as a -radian clockwise rotation about the segment/line determined by the center point and the top right vertex.

In the -dimensional Cartesian space , the -sphere has complete rotational symmetry in that its shape remains identical after any -radian rotation about any line . Historically, this fact led some ancient civilizations to consider the circle and/or the sphere to be divine (Weyl 1982).

In addition to being a well-studied concept mathematically, rotational symmetry is also a far-reaching notion due to the prevalence of such symmetry among many naturally-occurring objects including snowflakes, crystals, and flowers.

This entry contributed by Christopher Stover

More things to try:

Stover, Christopher. "Rotational Symmetry." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RotationalSymmetry.html