The triskelion appearing on the Isle of Man flag. A triskelion or triskele (both from the Greek gr ''τρισκέλιον'' or grc ''τρισκελής'' for "three-legged" is a Symbol The flag of the Isle of Man shows a Triskelion, the Three Legs of Mann emblem in the centre of a red flag

Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) 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"). Symmetry 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 A triskelion or triskele (both from the Greek gr ''τρισκέλιον'' or grc ''τρισκελής'' for "three-legged" is a Symbol The Isle of Man (Ellan Vannin ˈɛlʲən ˈvanɪn or Mann (Mannin) is a self-governing Crown dependency, located in the Irish Sea at the geographical 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. Symmetry 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 A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Rotations are direct isometries, i. For the Mechanical engineering and Architecture usage see Isometric projection. e. isometries preserving orientation. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which Therefore a symmetry group of rotational symmetry is a subgroup of E⁺(m) (see Euclidean group). The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional

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). In Geometry, a translation "slides" an object by a vector a: T a (p = p + a 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. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T For m=3 this is the rotation group. This article is about rotations in three-dimensional Euclidean space

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. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional For chiral objects it is the same as the full symmetry group. 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

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 homosexual system is equivalent to the angular momentum conservation law. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position See also Rotational invariance. In Mathematics, a function defined on an Inner product space is said to have rotational invariance if its value does not change when arbitrary Rotations

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 C_n or simply "n". A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions The actual symmetry group is specified by the point or axis of symmetry, together with the n. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is For each point or axis of symmetry the abstract group type is cyclic group Z_n of order n. 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 Although for the latter also the notation C_n is used, the geometric and abstract C_n should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D. In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group

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

Examples without additional reflection symmetry:

• n = 2, 180°: the dyad ,quadrilaterals with this symmetry are the parallelograms; other examples: letters Z, N, S; apart from the colors: yin and yang
• n = 3, 120°: triad, triskelion, Borromean rings; sometimes the term trilateral symmetry is used;
• n = 4, 90°: tetrad , swastika
• n = 6, 60°: hexad , raelian symbol, new version

C_n is the rotation group of a regular n-sided polygon in 2D and of a regular n-sided pyramid in 3D. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In Chinese philosophy, the concept of yin and yang ( is used to describe how seemingly opposing forces are bound together intertwined and interdependent in the A triskelion or triskele (both from the Greek gr ''τρισκέλιον'' or grc ''τρισκελής'' for "three-legged" is a Symbol In Mathematics, the Borromean rings consist of three topological Circles which are linked and form a Brunnian link, i The swastika (from Sanskrit: svástika sa स्वस्तिक Hindu IS CORRECT if 'ि' is positioned incorrectly see -->) is Raëlism or Raëlian Church consists of the practitioners of a UFO religion founded by a former French sports-car journalist and test driver named Claude In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit A pyramid is a Building where the upper surfaces are triangular and converge on one point

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°. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller. A propeller is essentially a type of fan which transmits power by converting Rotational motion into Thrust for propulsion of a vehicle such as an

Examples

C2
Nederlandse Spoorwegen (Dutch Railways) logo Double Pendulum fractal
C3
Roundabout traffic sign Snoldelev Stone's interlocked drinking horns design
C4
Decorative Hindu form of the swastika

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 D_n of order 2n (n≥2). Nederlandse Spoorwegen ( Dutch Railways) or NS, is the principal passenger Railway operating company in the Netherlands. In Horology, a double pendulum is a system of two simple Pendulums on a common mounting which move in anti-phase A roundabout is a type of Road junction at which Traffic enters a one-way stream around a central island Most countries post signage known as traffic signs or road signs, at the side of Roads to The 9th century Runestone at Snoldelev, Ramsø, Denmark, is decorated with a design of three Drinking horns interlocking as incomplete A drinking horn was a Drinking vessel formerly common in some parts of the world and notably in Northern Europe. A Hindu ( Devanagari: हिन्दू is an adherent of the philosophies and scriptures of Hinduism, a set of religious, Philosophical The swastika (from Sanskrit: svástika sa स्वस्तिक Hindu IS CORRECT if 'ि' is positioned incorrectly see -->) is In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections This is the rotation group of a regular prism, or regular bipyramid. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the Although the same notation is used, the geometric and abstract D_n should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections
• 4×3-fold and 3×2-fold axes: the rotation group T of order 12 of a regular tetrahedron. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. The group is isomorphic to alternating group A₄. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, an alternating group is the group of Even permutations of a Finite set.
• 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. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. An octahedron (plural octahedra is a Polyhedron with eight faces The group is isomorphic to symmetric group S₄. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying
• 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ 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. In Geometry, a Platonic solid is a convex Regular polyhedron. 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. Circular Symmetry in Mathematical physics applies to a 2-dimensional field which can be expressed as a function of distance from a central point only 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. Coordinates are numbers which describe the location of points in a plane or in space Coordinates are numbers which describe the location of points in a plane or in space The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. In Geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space Axisymmetric or axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry. In Grammar, an adjective is a word whose main syntactic role is to modify a Noun or Pronoun, giving more information about the 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. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. g. the duocylinder and various regular duoprisms. The duocylinder, or double cylinder, is a geometric object embedded in 4- Dimensional Euclidean space, defined as the Cartesian product of Nomenclature Four-dimensional duoprisms are considered to be prismatic Polychora.

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. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

Rotational symmetry with translational symmetry

Arrangement within a primitive cell of 2- and 4-fold rotocenters. In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding A fundamental domain is indicated in yellow. 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

2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a A frieze group is a mathematical concept to classify designs on Two-dimensional surfaces which are repetitive in one direction based on the symmetries in the pattern There are two rotocenters per primitive cell. In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding

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. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of
• 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. Plane tilings by Regular polygons have been widely used since antiquity
• p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides
• p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice. Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees

• 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

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. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides 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.

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. In Geometry, the bisected hexagonal tiling is a tiling of the Euclidean plane Rectangular line grids in three orientations can be distinguished.

See also

External links

• Rotational Symmetry Examples from Math Is Fun