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Rotational Symmetry


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

For d arbitrary, a geometric object in R^d 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 n is called the degree or the order of the symmetry.

Rotational symmetry of degree n corresponds to a plane figure being the same when rotated by 360/n degrees, or by 2pi/n radians.

RotationalSymmetryPentagon

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 alpha=2pin/5 radians, n=0,1,2,3,4, 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 n. In the case of the regular planar n-gon, the collection of all such symmetries is a group denoted by C_n, is isomorphic to the cyclic group Z/nZ of integers modulo n, and is a proper subgroup of the dihedral group D_n 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 360/1=360 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 n is defined by classifying the results of rotating a figure about a line l rather than about a point (Weyl 1982). In particular, such sources define a figure to have rotational symmetry of order n if the figure which remains identical after a 2pi/n-radian rotation about l (which is called the axis of rotation). These two perspectives yield the same result, however; for example, in the figure above, the 2pi/5-radian clockwise rotation of the pentagon about its center point can equivalently be viewed as a 2pi/5-radian clockwise rotation about the segment/line determined by the center point and the top right vertex.

In the (d+1)-dimensional Cartesian space R^(d+1), the d-sphere S^d has complete rotational symmetry in that its shape remains identical after any alpha-radian rotation about any line l. 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.


See also

Affine Transformation, Automorphism Group, Cyclic Group, Dihedral Group, Reflection, Rotation, Symmetry, Symmetry Operation

This entry contributed by Christopher Stover

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References

Weyl, H. Symmetry. Princeton, NJ: Princeton University Press, 1982.

Cite this as:

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

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