Encyclopedia
The
symmetry group of an object 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.
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 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.
The symmetry group is sometimes also called
full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries 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.
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 by choosing the origin to be a fixed point. The proper symmetry group is a subgroup of the special orthogonal group SO then, and therefore also called
rotation group of the figure.
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, 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 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 , 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 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. 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, also denoted by D∞ .
- the group generated by all translations ; 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.
See also symmetry groups 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,... 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 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.
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.
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 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 consisting of all rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex number
...
s of
absolute value 1. 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 .
- the orthogonal group O 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 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 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. 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 .
See
point groups in three dimensions.
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,
has cylindrical symmetry with respect to the axis if and only if and have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry if and only if .
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 themselves. See .
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 . 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 and its proof.
See also
External links
- - form the first parts of the 7 infinite series and 5 of the 7 separate 3D point groups
