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.
