Symmetry group
The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept.In Euclidean geometry, discrete symmetry groups come in two types: finite point groups, which include only rotations and reflections, and infinite lattice groups, which also include translations and glide reflections. There are also continuous symmetry groups, which are Lie groups.
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2 Three dimensions 3 Symmetry groups in general 4 Related topics |
The two simplest point groups in 2-D space are the trivial group C1, where no symmetry operations leave the object unchanged, and the group containing only the identity and reflection about a particular line, D1. The other point groups form two infinite series, called Cn and Dn: the cyclic groups and the dihedral groups. The former is generated by a rotation by 2π/n radians about a particular point, and the latter by such a rotation together with a reflection about a line that runs through that point.
The situation in 3-D is more complicated, since it is possible to have multiple rotation axes in a point group. First, of course, there is the trivial group, and then there are three groups of order 2, called Cs, Ci, and C2. These have the single symmetry operation of reflection about a plane, about a point, and about a line (equivalent to a rotation of π), respectively.
The last of these is the first of the uniaxial groups Cn, which are generated by a single rotation of angle 2π/n. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group Cnh, or a set of n mirror planes containing the axis, giving the group Cnv. If both are added, there intersections give n axes of rotation through π, so the group is no longer uniaxial.
This new group is called Dnh, and its subgroup of rotations is Dn. There is one more group in this family, called Dnd, which has mirror planes containing the main rotation axis but located halfway between the others, so the perpendicular plane is not there. Dnh and Dnd are the symmetry groups for regular prisms and antiprisms, respectively.
There is one more group in this family to mention, called Sn. This group is generated by an improper rotation of angle 2π/n - that is, a rotation followed by a reflection about a plane perpendicular to its axis. For n even, the rotation and reflection are generated, so this becomes the same as Cnh, but it remains distinct for n odd.
The remaining point groups are said to be of very high or polyhedral symmetry because they have more than one rotation axis of order greater than 2. Using Cn to denote an axis of rotation through 2π/n and Sn to denote an axis of improper rotation through the same, the groups are:
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.Two dimensions
Examples
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C1 D1 C2 D4
Groups including translation in a single direction are called frieze groups. There are seventeen 2-D lattice groups including translation in multiple directions, called wallpaper groups.Three dimensions
Symmetry groups in general