Symmetry group

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 an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in more general contexts as expanded below.

Introduction

The "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern. 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 physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a group action on objects in it.

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) that 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).

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 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 therefore also called rotation group of the figure.

A discrete symmetry group is a symmetry group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. The number of elements in the group may be either finite or infinite.

Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion – they are 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 displacements and 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. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.

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. For example:

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

When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also those involved in continuous symmetries, but 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 one dimension 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 additive group of the integers
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).
the group generated by all translations (isomorphic with the additive group of the real numbers R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform one-dimensional 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).

Two dimensions

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

cyclic groups C₁, C₂, C₃, C₄, ... where C_n consists of all rotations about a fixed point by multiples of the angle 360°/n
dihedral groups D₁, D₂, D₃, D₄, ..., where D_n (of order 2n) consists of the rotations in C_n 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.

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.

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.

The remaining isometry groups in two dimensions 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. It is the proper symmetry group of a circle and the continuous equivalent of C_n. There is no geometric figure that has as full symmetry group the circle group, but for a vector field it may apply (see the three-dimensional 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.

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 one dimension, 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 three-dimensional 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 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. However, for vector fields it does not: in cylindrical coordinates with respect to some axis, $\mathbf {A} =A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}$ has cylindrical symmetry with respect to the axis if and only if $A_{\rho },A_{\phi },$ and $A_{z}$ have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry if and only if $A_{\phi }=0$ .

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. 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. 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.

For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. 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.

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.

Compare Lagrange's theorem (group theory) and its proof.

External links

Weisstein, Eric W. "Symmetry Group". MathWorld.

Weisstein, Eric W. "Tetrahedral Group". MathWorld.

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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