The triskelion appearing on the Isle of Man flag.

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. cropped from Image:Man flag large. ... cropped from Image:Man flag large. ... This article needs additional references or sources for verification. ... Flag ratio: 1:2 The flag of the Isle of Man shows a triskelion, the Three Legs of Man emblem, in the centre of a red flag. ... This article is about rotation as a movement of a physical body. ... Sphere symmetry group o. ... This article needs additional references or sources for verification. ...

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). Sphere symmetry group o. ... This article is about rotation as a movement of a physical body. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... The symmetry group of an object (e. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

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. In geometry, a translation slides an object by a vector a: Ta(p) = p + a. ...

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 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. ... In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, The definition can be given for matrices with entries from any field, but the most common case is the one of matrices with real entries, and only that case will be considered in the... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

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. In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ... 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. ...

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. Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... This gyroscope remains upright while spinning due to its angular momentum. ... This article or section does not cite its references or sources. ...

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". 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 Z_n of order n. 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. A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of point groups. ... The symmetry group of an object (e. ... 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 of... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

The fundamental domain is a sector of 360°/n. In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...

Examples without additional reflection symmetry: Figures with the axes of symmetry drawn in. ...

• 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. Etymology: Late Latin dyad-, dyas, from Greek, from dyo The word dyad has a number of uses: A dyad (general) pair, consisting of two parts. ... This article is about the geometric shape. ... A parallelogram. ... Japanese name Kanji: Hiragana: Korean name Hangul: Hanja: Vietnamese name Quốc ngữ: Chữ nôm: Hán tá»±: The Taijitu of Zhou Dun-yi In Chinese philosophy, yin and yang (simplified Chinese: ; traditional Chinese: ; pinyin: ) are generalized descriptions of the antitheses or mutual correlations in human perceptions of phenomena... Triad (Traditional Chinese: ; Simplified Chinese: ; Pinyin: ; literally Triad Society) or (Traditional Chinese: ; Simplified Chinese: ; Pinyin: ; literally Black Society, a general term for criminal organizations) is a term that describes many branches of Chinese underground society and/or organizations based in Hong Kong and Macau and also operating in Taiwan, mainland... This article needs additional references or sources for verification. ... In mathematics, the Borromean rings consist of three topological circles which are linked despite the fact that no two of them are linked, i. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... This article is about the symbol. ... Raels first published book, the basis of the Raelian movement Raëlism is the belief system promoted by the Raëlian Movement, a religious organization which believes that scientifically advanced extraterrestrials known as the Elohim (one of the words used to refer to God in the Torah) created life... Look up polygon in Wiktionary, the free dictionary. ... For other meanings, see pyramid (disambiguation). ...

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 integers, is the largest positive integer that divides both numbers without remainder. ...

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller. For other uses, see Propeller (disambiguation). ...

Examples

C2		C3
Roundabout traffic sign Image File history File links Liikenneympyrä_166. ... A roundabout is a type of road junction at which traffic enters a one-way stream around a central island. ... Unused traffic signs in Austria Most countries post signage, known as traffic signs or road signs, at the side of roads to impart information to road users. ... Snoldelev Stone's interlocked drinking horns design Image File history File links No higher resolution available. ... The Viking Age (Younger Futhark) runestone at Snoldelev, Ramsø, Denmark, dated to ca. ... A drinking horn was a drinking vessel formerly common in some parts of the world. ...
C4		Mixed
===Multiple symmetry axes through the same point=== For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities: ===Rotational symmetry with respect to any angle=== ====Geometry, architecture and furniture==== ===Rotational symmetry with translational symmetry=== *6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice. == See also == == External links ==