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Encyclopedia > Symmetry group

The symmetry group of an object (e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant), with composition as the operation. It is a subgroup of the isometry group of the space concerned. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, an invariant is something that does not change under a set of transformations. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

(If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts, see below.) In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...

The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations) under which the figure is invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations and compositions of these) which leave the figure invariant is called its rotation group or proper symmetry group. The rotation group of an object is equal to its full symmetry group iff the object is chiral. Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... 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. ...

Any symmetry group whose elements have a common fixed point can be represented as a subgroup of orthogonal group O(n) (by choosing the origin to be a fixed point). This is true for all finite symmetry groups, and also for the symmetry groups of bounded figures. In this case the proper symmetry group of an object is the intersection of its full symmetry group and the rotation group SO(n) of 3D space itself. In mathematics, a fixed point of a function is a point that is mapped to itself by the function. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a 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 mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...

Discrete symmetry groups come in two types: finite point groups, which include only rotations, reflections, and combinations - they are in fact just the finite subgroups of O(n), and infinite lattice groups, which also include translations and possibly glide reflections. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. In mathematics, a discrete group is a group G equipped with the discrete topology. ... A cherry lattice pastry A mathematical lattice is a type of partially ordered set. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... 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 mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of Iso(Rn) (where Iso(Rn) is the isometry group of Rn and two subgroups H1, H2 of a group G are conjugate, if there exists gG such that H1=g-1H2g ). For example: In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

• two 3D figures have mirror symmetry, but with respect to a different mirror plane
• two 3D figures have 3-fold rotational symmetry, but with respect to a different axis
• two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction

Sometimes a broader concept of "same symmetry type" is used, resulting in e.g. 17 wallpaper groups.

## Contents

Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:

• cyclic groups C1, C2, C3, C4,... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n
• dihedral groups D1, D2, D3, D4,... where Dn (of order 2n) consists of the rotations in Cn together with reflections in n axes that pass through the fixed point.

C1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C2 is the symmetry group of the letter Z, C3 that of a triskelion, C4 of a swastika, and C5, C6 etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ... The smallest non-Abelian group has 6 elements. ... Some elementary examples of groups in mathematics are given on Group (mathematics). ... The armoured triskelion on the flag of the Isle of Man Triskelion (or triskele, from Greek Ï„ÏÎ¹ÏƒÎºÎµÎ»Î·Ï‚ three-legged) is a symbol consisting of three bent human legs, or, more generally, three interlocked spirals, or any similar symbol with three protrusions exhibiting a symmetry of the cyclic group C3. ... The swastika (å) is an equilateral cross with its arms bent at right angles either clockwise or anticlockwise. ...

D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D3, D4 etc. are the symmetry groups of the regular polygons. In biology, bilateral symmetry is a characteristic of multicellular organisms, particularly animals. ... In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is the group C2 Ã— C2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Providing the figure is bounded and topologically closed (so that the group is a complete point group) the only other possibility is the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. The closure condition here is a natural one for subsets of the plane that can be considered "figures", as it excludes non-drawable sets such as the set of all points on the unit circle with rational coordinates. The symmetry group of this set includes some, but not all, arbitrarily small rotations. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... 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. ... Illustration of a unit circle. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

The proper symmetry group of a circle is the circle group S1, i.e. the multiplicative group of complex numbers of absolute value 1. It is the continuous equivalent of Cn, but there is no figure which has as full symmetry group the circle group. O(2) is also called Dih(S1) as it is the generalized dihedral group of S1. In mathematics, the circle group is the group of all complex numbers on the unit circle under multiplication. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ...

For non-bounded figures, the symmetry group can include translations, so that the seventeen wallpaper groups and seven frieze groups are possibilities. Example of a Persian design with wallpaper group p6m A wallpaper group (or plane crystallographic group) is a mathematical device used to describe and classify repetitive designs on two-dimensional surfaces, such as walls. ... A frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). ...

## Three dimensions

Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others). The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ... In crystallography, a crystallographic point group or crystal class is a set of symmetry operations that leave a point fixed, like rotations or reflections, which leave the crystal unchanged. ...

See point groups in three dimensions. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ...

The continuous symmetry groups with a fixed point include those of:

• cylindrical symmetry with or without a symmetry plane perpendicular to the axis
• spherical symmetry

## Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...

## Related topics

• Overview of the 32 crystallographic point groups - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups Results from FactBites:

 Symmetry group - Wikipedia, the free encyclopedia (1546 words) Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. This is the symmetry group of a circle.
 Symmetry - Wikipedia, the free encyclopedia (2564 words) For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, and ornamentation; many facades adhere to bilateral symmetry. Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord.
More results at FactBites »

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