- This article is about the abstract algebraic structures. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules For other meanings, see Symmetry group (disambiguation).
The symmetry group of an object (image, signal, etc. 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 An image (from Latin imago) or picture is an artifact usually two-dimensional that has a similar appearance to some subject &mdashusually In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity , e. g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Mathematics, a composite function represents the application of one function to the results of another It is a subgroup of the isometry group of the space concerned. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, the isometry group of a Metric space is the set of all isometries from the metric space onto itself with the Function composition
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Introduction
(If not stated otherwise, we consider symmetry groups in Euclidean geometry here, but the concept may also be studied in wider contexts, see below. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. )
The "objects" may be geometric figures, images and patterns, such as a wallpaper pattern. A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the The definition can be made more precise by specifying what is meant by image or pattern, e. g. a function of position with values in a set of colors. For symmetry of e. g. 3D bodies one may also want to take physical composition into account. The group of isometries of space induces a group action on objects in it. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.
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. In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or The subgroup of orientation-preserving isometries (i. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of e. translations, rotations and compositions of these) which leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant). ↔ 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
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. In Mathematics, a fixed point (sometimes shortened to fixpoint) 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 In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n The proper symmetry group is a subgroup of the special orthogonal group SO(n) then, and therefore also called rotation group of the figure.
Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion - they are in fact just the finite subgroups of O(n), (2) infinite lattice groups, which include only translations, and (3) infinite space groups which combines elements of both previous types, and may also include extra transformations like screw axis and glide reflection. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation 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, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetries as motions as opposed to e In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional
Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rn), where two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1=g-1H2g. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class For example:
- 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.
When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. This excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.
One dimension
The isometry groups in 1D where for all points the set of images under the isometries is topologically closed are:
- the trivial group C1
- the groups of two elements generated by a reflection in a point; they are isomorphic with C2
- the infinite discrete groups generated by a translation; they are isomorphic with Z
- the infinite discrete groups generated by a translation and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D∞ (which is a semidirect product of Z and C2). In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can
- the group generated by all translations (isomorphic with R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform 1D vector field has this symmetry group.
- the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group of R, Dih(R). In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections
See also symmetry groups in one dimension. A one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension
Two dimensions
Up to conjugacy the discrete point groups in 2 dimensional space are the following classes:
- cyclic groups C1, C2, C3, C4,. 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 . . where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n
- dihedral groups D1, D2, D3, D4,. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections The smallest Non-abelian group has 6 elements It is a Dihedral group with notation D 3 or D 6 Some elementary examples of groups in Mathematics are given on Group (mathematics. . . 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. A triskelion or triskele (both from the Greek gr ''τρισκέλιον'' or grc ''τρισκελής'' for "three-legged" is a Symbol The swastika (from Sanskrit: svástika sa स्वस्तिक Hindu IS CORRECT if 'ि' is positioned incorrectly see -->) is are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four.
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. "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D3, D4 etc. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 are the symmetry groups of the regular polygons. General properties These properties apply to both convex and star regular polygons
The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors. For information on degrees of freedom in other sciences see Degrees of freedom.
The remaining isometry groups in 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are:
- the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex numbers of absolute value 1. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. It is the proper symmetry group of a circle and the continuous equivalent of Cn. There is no figure which has as full symmetry group the circle group, but for a vector field it may apply (see the 3D case below).
- 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. It is also called Dih(S1) as it is the generalized dihedral group of S1. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections
For non-bounded figures, the additional isometry groups can include translations; the closed ones are:
- the 7 frieze groups
- the 17 wallpaper groups
- for each of the symmetry groups in 1D, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction
- ditto with also reflections in a line in the first direction
Three dimensions
Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. 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 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 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 was based on the observation that the Rotational symmetries of a Crystal are usually limited to
See point groups in three dimensions. 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 continuous symmetry groups with a fixed point include those of:
- cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle
- cylindrical symmetry with a symmetry plane perpendicular to the axis
- spherical symmetry
For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. A bottle is a container with a neck that is narrower than the body and a "mouth In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point However, for vector fields it does not: in cylindrical coordinates with respect to some axis, 
has cylindrical symmetry with respect to the axis if and only if Aρ,Aφ, and Az have this symmetry, i. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually ↔ e. , they do not depend on φ. Additionally there is reflectional symmetry if and only if Aφ = 0.
For spherical symmetry there is no such distinction, it implies planes of reflection.
The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i See also subgroups of the Euclidean group. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional
Symmetry groups in general
- See also: Automorphism
In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. In Mathematics, a structure on a set, or more generally a type, consists of additional Mathematical objects that in some manner attach to the 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. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen
For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. A finite geometry is any geometric system that has only a finite number of points. They do this by preserving families of point-sets rather than point-sets (or "objects") themselves.
Like above, the group of automorphisms of space induces a group action on objects in it. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.
For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same (here "the same" does not mean something like e. ↔ g. "the same up to translation and rotation", but it means "exactly the same"). Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure.
There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second.
In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure.
Examples:
- Isometries of the Euclidean plane, the figure is a rectangle: there are infinitely many equivalence classes; each contains 4 isometries.
- The space is a cube with Euclidean metric; the figures include cubes of the same size as the space, with colors or patterns on the faces; the automorphisms of the space are the 48 isometries; the figure is a cube of which one face has a different color; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.
Compare Lagrange's theorem (group theory) and its proof. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of
See also
- symmetric group
- symmetry
- symmetry groups in one dimension
- permutation group
- fixed points of isometry groups in Euclidean space
- Euclidean plane isometry
- group action
- point group
- crystal system
- space group
External links
- Eric W. Weisstein, Symmetry Group at MathWorld. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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 one-dimensional symmetry group is a mathematical group that describe symmetries in one dimension In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation A fixed point of an isometry group is a point that is a fixed point for every Isometry in the group In Geometry, a Euclidean plane isometry is an Isometry of the Euclidean plane, or more informally a way of transforming the plane that preserves geometrical In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Mathematics, a point group is a group of geometric symmetries ( isometries) leaving a point fixed A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
- Eric W. Weisstein, Tetrahedral Group at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
- 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
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