![]() Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E( n) (the isometry group of R n). An example is O(3), the symmetry group of a sphere. There are also continuous symmetry groups ( Lie groups), which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. All finite symmetry groups are discrete.ĭiscrete 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. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete set. In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit 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. An object is chiral when it has no orientation-reversing symmetries, so that its proper symmetry group is equal to its full symmetry group.Īny 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 subgroup of orientation-preserving symmetries (translations, rotations, and compositions of these) is called its proper symmetry group. 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. We say X is invariant under such a mapping, and the mapping is a symmetry of X. (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). For symmetry of physical objects, one may also take their physical composition as part of the pattern. We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. ![]() This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. A frequent notation for the symmetry group of an object X is G = Sym( X).įor an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. 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. 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. The twelve rotations form the rotation (symmetry) group of the figure. ![]() These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (pink and orange arrows) rotations that permute the tetrahedron through the positions. ( December 2017) ( Learn how and when to remove this template message)Ī regular tetrahedron is invariant under twelve distinct rotations (if the identity transformation is included as a trivial rotation and reflections are excluded). ![]() ![]() Please help to improve this article by introducing more precise citations. This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. ![]()
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