Symmetry group

Last updated
A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format, along with the 180deg edge (blue arrows) and 120deg vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure. Tetrahedral group 2.svg
A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

Contents

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

Introduction

We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, a function of position with values in a set of colors or substances; as a vector field; or as a more general function on the object.) The group of isometries of space induces a group action on objects in it, and the symmetry group Sym(X) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X.

The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular X to itself. The subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. An object is chiral when it has no orientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.

Any symmetry group whose elements have a common fixed point, which is true if the group is finite or the figure is bounded, can be represented as a subgroup of the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is called the rotation group of the figure.

In a discrete symmetry group , the points symmetric to a given point do not accumulate toward a limit point. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set. All finite symmetry groups are discrete.

Discrete symmetry groups come in three types: (1) finite point groups , which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections. There are also continuous symmetry groups (Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is O(3), the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of Rn).

Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H1, H2 are related by H1 = g−1H2g for some g in E(n). For example:

In the following sections, we only consider isometry groups whose orbits are topologically closed, including all discrete and continuous isometry groups. However, this excludes for example the 1D group of translations by a rational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.

One dimension

The isometry groups in one dimension are:

See also symmetry groups in one dimension.

Two dimensions

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

C1 is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, 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.

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. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes.

D3, D4 etc. are the symmetry groups of the regular polygons.

Within each of these symmetry types, there are 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.

The remaining isometry groups in two dimensions with a fixed point are:

Non-bounded figures may have isometry groups including translations; these are:

Three dimensions

Up to conjugacy the set of three-dimensional point groups consists of 7 infinite series, and 7 other individual groups. In crystallography, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6). This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 individual groups from the 7 series, and 5 of the 7 other individuals).

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

For objects with scalar field patterns, the cylindrical symmetry implies vertical reflection symmetry as well. However, this is not true for vector field patterns: for example, in cylindrical coordinates with respect to some axis, the vector field has cylindrical symmetry with respect to the axis whenever and have this symmetry (no dependence on ); and it has reflectional symmetry only when .

For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

Symmetry groups in general

In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the Erlangen programme.

For example, objects in a hyperbolic non-Euclidean geometry have Fuchsian symmetry groups, which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of Escher.) Similarly, automorphism groups of finite geometries preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space.

Another example of a symmetry group is that of a combinatorial graph: a graph symmetry is a permutation of the vertices which takes edges to edges. Any finitely presented group is the symmetry group of its Cayley graph; the free group is the symmetry group of an infinite tree graph.

Group structure in terms of symmetries

Cayley's theorem states that any abstract group is a subgroup of the permutations of some set X, and so can be considered as the symmetry group of X with some extra structure. In addition, many abstract features of the group (defined purely in terms of the group operation) can be interpreted in terms of symmetries.

For example, let G = Sym(X) be the finite symmetry group of a figure X in a Euclidean space, and let HG be a subgroup. Then H can be interpreted as the symmetry group of X+, a "decorated" version of X. Such a decoration may be constructed as follows. Add some patterns such as arrows or colors to X so as to break all symmetry, obtaining a figure X# with Sym(X#) = {1}, the trivial subgroup; that is, gX#X# for all non-trivial gG. Now we get:

Normal subgroups may also be characterized in this framework. The symmetry group of the translation gX+ is the conjugate subgroup gHg1. Thus H is normal whenever:

that is, whenever the decoration of X+ may be drawn in any orientation, with respect to any side or feature of X, and still yield the same symmetry group gHg1 = H.

As an example, consider the dihedral group G = D3 = Sym(X), where X is an equilateral triangle. We may decorate this with an arrow on one edge, obtaining an asymmetric figure X#. Letting τG be the reflection of the arrowed edge, the composite figure X+ = X#τX# has a bidirectional arrow on that edge, and its symmetry group is H = {1, τ}. This subgroup is not normal, since gX+ may have the bi-arrow on a different edge, giving a different reflection symmetry group.

However, letting H = {1, ρ, ρ2} D3 be the cyclic subgroup generated by a rotation, the decorated figure X+ consists of a 3-cycle of arrows with consistent orientation. Then H is normal, since drawing such a cycle with either orientation yields the same symmetry group H.

See also

Further reading

Related Research Articles

Group theory Branch of mathematics that studies the properties of groups

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Dihedral group Group of symmetries of a regular polygon

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

Frieze group Type of symmetry group

In mathematics, a frieze or frieze pattern is a design on a two-dimensional surface that is repetitive in one direction. Such patterns occur frequently in architecture and decorative art. A frieze group is the set of symmetries of a frieze pattern, specifically the set of isometries of the pattern, that is geometric transformations built from rigid motions and reflections that preserve the pattern. The mathematical study of frieze patterns reveals that they can be classified into seven types according to their symmetries.

Glide reflection Symmetry operation combining reflection across and translation along an axis

In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection and translation can look different from the starting configuration, so objects with glide symmetry are in general, not symmetrical under reflection alone. In group theory, the glide plane is classified as a type of opposite isometry of the Euclidean plane

Discrete group

In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H ; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.

Euclidean group Isometry group of Euclidean space

In mathematics, a Euclidean group is the group of (Euclidean) isometries of an Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n).

Rotational symmetry Symmetry (something looking the same) under rotation

Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Point group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O(d). Point groups can be realized as sets of orthogonal matrices M that transform point x into point y:

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 properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

A one-dimensional symmetry group is a mathematical group that describes symmetries in one dimension (1D).

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections, it is a continuous group, not a discrete group, and is generally considered separately.

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 of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries.

Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.

Tetrahedral symmetry 3D symmetry group

A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

In geometry, orbifold notation is a system, invented by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.

In a group, the conjugate by g of h is ghg−1.

Point groups in two dimensions

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

Infinite dihedral group

In mathematics, the infinite dihedral group Dih is an infinite group with properties analogous to those of the finite dihedral groups.

In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group O(2). Dihedral groups play an important role in group theory, geometry, and chemistry.

Symmetry (geometry)

In geometry, an object has symmetry if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.