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Information about Rotational Symmetry

Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag (see opposite) has three rotational symmetries (or "a threefold rotational symmetry"). More examples may be seen below.

Formal treatment

Formally, 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 vertex implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

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 m=3 this is the rotation group.

In another meaning 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, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also Rotational invariance.

n-fold rotational symmetry

Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of 360°/n (180°, 120°, 90°, 72°, 60°, 51 3/7 °, etc.) does not change the object. Note that "1-fold" symmetry is no symmetry, and "2-fold" is the simplest symmetry, so it does mean "more than basic".

The notation for n-fold symmetry is Cn or simply "n". The actual symmetry group is specified by the point or axis of symmetry, together with the n. For each point or axis of symmetry the abstract group type is cyclic group Zn of order n. Although for the latter also the notation Cn is used, the geometric and abstract Cn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D.

The fundamental domain is a sector of 360°/n.

Examples without additional reflection symmetry: Cn is the rotation group of a regular n-sided polygon in 2D and of a regular n-sided pyramid in 3D.

If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller.

Examples

C2 C3

Nederlandse Spoorwegen (Dutch Railways) logo

Double Pendulum fractal

Roundabout traffic sign

Snoldelev Stone's interlocked drinking horns design
 
C4 Mixed

C4 rotational symmetry in the circular area of the Syrian Social Nationalist Party flag

Decorative Hindu form of the swastika

Raëlian symbol, before and since 1991: 6-fold; 6/4-fold; 2-fold rotational symmetries (left; right; both)

Multiple symmetry axes through the same point

For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:
  • In addition to an n-fold axis, n perpendicular 2-fold axes: the dihedral groups Dn of order 2n (n≥2). This is the rotation group of a regular prism, or regular bipyramid. Although the same notation is used, the geometric and abstract Dn should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D.
  • 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron. The group is isomorphic to alternating group A4.
  • 3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron. The group is isomorphic to symmetric group S4.
  • 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group A5. The group contains 10 versions of D3 and 6 versions of D5 (rotational symmetries like prisms and antiprisms).
In the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.

Rotational symmetry with respect to any angle

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. 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.

Geometry, architecture and furniture

Rotational symmetry is a perfectly symmetrical shape wherein a two dimensional object is necessarily circular, and a three dimensional object may be considered as a stack of discs of differing radii.

Rotational symmetry with translational symmetry

Enlarge picture
Arrangement within a primitive cell of 2- and 4-fold rotocenters. A fundamental domain is indicated in yellow.
2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. There are two rotocenters per primitive cell.

Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell:
  • p2 (2222): 4×2-fold; rotation group of a parallelogrammic, rectangular, and rhombic lattice.
  • p3 (333): 3×3-fold; not the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored.
  • p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice.
  • p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice.
  • 2-fold rotocenters (including possible 4-fold and 6-fold), if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not apply.
  • 3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor
Enlarge picture
Arrangement within a primitive cell of 2-, 3-, and 6-fold rotocenters, alone or in combination (consider the 6-fold symbol as a combination of a 2- and a 3-fold symbol); in the case of 2-fold symmetry only, the shape of the parallelogram can be different. For the case p6, a fundamental domain is indicated in yellow.
  • 4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor
  • 6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.
Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.

3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is 2√3 times their distance.

Enlarge picture
Hexakis triangular tiling, an example of p6 (with colors) and p6m (without); the lines are reflection axes if colors are ignored, and a special kind of symmetry axis if colors are not ignored: reflection reverts the colors. Rectangular line grids in three orientations can be distinguished.

See also



External links

This article is about rotation as a movement of a physical body. For other uses, see Rotation (disambiguation).
A rotation is a movement of an object in a circular motion.
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Symmetry in common usage generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection.
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triskelion or triskele (both from the Greek, τρισκέλιον) or τρισκελής
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Ellan Vannin
Isle of Man


Flag Coat of arms
Motto
Quocunque Jeceris Stabit
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Symmetry in common usage generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection.
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This article is about rotation as a movement of a physical body. For other uses, see Rotation (disambiguation).
A rotation is a movement of an object in a circular motion.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent.
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See also orientation (rigid body).


In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which are "negatively" oriented.
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rotation (symmetry) group of the figure.]]

The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation.
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In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves.
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a: Ta(p) = p + a.

In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation.
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In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
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In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse:



An orthogonal matrix is a special orthogonal matrix if it has determinant +1:

Overview


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In mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R3 under the operation of composition.
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In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves.
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In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone.
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This article or section may be confusing or unclear for some readers.
Please [improve the article] or discuss this issue on the talk page. This article has been tagged since August 2007.
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angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.
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rotational invariance if its value does not change when arbitrary rotations are applied to its argument. For example, the function f(x,y) = x2 + y2 is invariant under rotations of the plane around the origin.
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rotation (symmetry) group of the figure.]]

The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation.
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In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power
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In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern. The set of orbits of the symmetry group define a partitioning of space.
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Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.

It is the most common type of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry.
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quadrilateral is a polygon with four sides or edges and four vertices or corners. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon
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In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.
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Yin and yang (Simplified Chinese: 阴阳; Traditional Chinese: 陰陽; Pinyin: yīnyáng
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triskelion or triskele (both from the Greek, τρισκέλιον) or τρισκελής
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In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.

Mathematical properties


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swastika (from Sanskrit svástika
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