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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.
If a figure is bounded, then all elements of its symmetry group have a common fixed point.
1 Two dimensions
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.
1.1 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.
2 Three dimensions
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 (or C1h), Ci, and C2. These have the single symmetry operation of reflection in a plane, in a point, and in 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 horizontal and vertical reflection planes are added, their intersections give n axes of rotation through π, so the group is no longer uniaxial. This new group is called Dnh. Its subgroup of rotations called Dn still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. There is one more group in this family, called Dnd (or Dnv), which has vertical mirror planes containing the main rotation axis but located halfway between the other 2-fold axes, so the perpendicular plane is not there. Dnh and Dnd are the symmetry groups for regular prisms and antiprisms, respectively. Dn is the symmetry group of a partially rotated prism.
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:
- T (tetrahedral). There are four C3 axes, directed through the corners of a cube, and three C2 axes, directed through the centers of its faces. There are no other symmetry operations, giving the group an order of 12. This group is isomorphic to A4, the alternating groupIn mathematics an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,. n is called the alternating group of degree n or the alternating group on n letters and denoted by A. For instance: {1234, 1342, 1 on 4 letters.
- Td. This group has the same rotation axes as T, but with six mirror planes, each containing a single C2 axis and four C3 axes. The C2 axes are now actually S4 axes. This group has order 24, and is the symmetry group for a regular tetrahedronA tetrahedron (plural: tetrahedra is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral," and is one of the Platonic solids. The area A. Td is isomorphic to S4, the symmetric groupIn mathematics, the symmetric group on a set X denoted by S or Sym X , is the group whose underlying set is the set of all bijective functions from X to X in which the group operation is that of composition of functions, i. two such functions f and g can on 4 letters.
- Th. This group has the same rotation axes as T, but with mirror planes, each containing two C2 axes and no C3 axes. The C3 axes become S6 axes, and a center of inversion appears. Again, group has order 24. Th is isomorphic to A4 × C2.
- O (octahedral). This group is similar to T, but the C2 axes are now C4 axes, and a new set of 12 C2 axes appear, directed towards the edges of the original cube. This group of order 24 is also isomorphic to S4.
- Oh. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group has order 48, is isomorphic to S4 × C2, and is the symmetry group of the cubeThree dimensions A cube (or hexahedron is a Platonic solid composed of six square faces, with three meeting at each vertex. The cube is a special kind of square prism, of rectangular parallelepiped and of triangular trapezohedron, and is dual to the octah and octahedronAn octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight faces each of which is an equilateral triangle four of which meet at each vertex. The regular octahedron is a special kind of tr.
- I, Ih (icosahedral) are the groups of symmetries of the icosahedronAn icosahedron [aiks'hidrn] noun (plural: -drons, -dra [-dr]) is a polyhedron having 20 faces. The faces of a regular icosahedron are equilateral triangles. Etymology 16th Century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], "icosa'hedra and the dodecahedronA dodecahedron is a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. It has twenty vertices and thirty edges. Its dual polyhedron is the icosahedron. Canonical coordinates for the vertices of a dodecahedron centered a. The group of proper rotations, I, is a normal subgroupIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t of index 2 in the full group of symmetries, with I having order 60 and Ih having order 120. The group I is isomorphic to A5, the alternating group on 5 letters, and Ih to A5 × C2....
