Editing Symmetry group (section)

==Introduction==
We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a [[wallpaper group|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 (mathematics)|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 reflection]]s and [[improper rotation]]s), 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 [[chirality (mathematics)|chiral]] when it has no [[Orientation (vector space)|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 (mathematics)|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 group|discrete symmetry group]]''', the points symmetric to a given point do not accumulate toward a [[limit point]]. That is, every [[Orbit (group theory)|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 group]]s''', which include only rotations, reflections, inversions and [[Improper rotation|rotoinversions]] – i.e., the finite subgroups of O(''n''); (2) infinite '''[[Lattice (group)|lattice]] groups''', which include only translations; and (3) infinite '''[[space group]]s''' containing elements of both previous types, and perhaps also extra transformations like [[screw displacement]]s and glide reflections. There are also [[continuous symmetry]] groups ([[Lie group]]s), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. An example is [[orthogonal group|O(3)]], the symmetry group of a sphere. Symmetry groups of Euclidean objects may be completely classified as the [[Euclidean group#Subgroups|subgroups of the Euclidean group]] E(''n'') (the isometry group of '''R'''<sup>''n''</sup>).

Two geometric figures have the same ''symmetry type'' when their symmetry groups are ''[[Conjugacy class#Conjugacy of subgroups and general subsets|conjugate]]'' subgroups of the Euclidean group: that is, when the subgroups ''H''<sub>1</sub>, ''H''<sub>2</sub> are related by {{nowrap|1=''H''<sub>1</sub> = ''g''<sup>−1</sup>''H''<sub>2</sub>''g''}} for some ''g'' in E(''n''). 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.

In the following sections, we only consider isometry groups whose [[Orbit (group theory)|orbits]] are [[Closed (topology)|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.