Rotational symmetry, also called rotation symmetry, is the property of a figure remaining unchanged under a nontrivial rotation.
Abstractly, a spatial configuration 
is said to possess rotational symmetry if 
contains a rotation other
than the identity map. 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 Euclidean
space.
For a planar figure in 
, the rotations about a fixed
point that leave the figure unchanged form its rotational
symmetry group. If this group is finite, its number of
elements 
is called the order of the
rotational symmetry.
For a plane figure with a finite rotational symmetry group
of order 
, the smallest positive angle of rotational symmetry
is 
degrees, or 
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 planar regular
polygon with 
polygon edges has rotational symmetry of order 
. For such a planar regular
polygon, 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.
A figure whose rotational symmetry group is trivial has order 1; it returns to itself only after a full rotation of 360 degrees. Such a figure has no nontrivial rotational symmetry, even if it has reflection symmetry. The simplest nontrivial rotational symmetry has 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 
about 
if 
radians is the smallest positive angle of rotation
about 
that leaves the figure unchanged. The line 
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 line through the center perpendicular to the plane
of the pentagon.
In the 
-dimensional Euclidean space 
, the 
-sphere 
has complete rotational symmetry
in that its shape remains identical under every rotation
fixing its center. For a sphere
in three dimensions, this includes every rotation
about a line through its center.
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.
