Symmetry group information

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...

corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane...

known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many...

mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest...

molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a...

Symmetry (from Ancient Greek συμμετρία (summetría) 'agreement in dimensions, due proportion, arrangement') in everyday life refers to a sense of harmonious...

preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E +(m) (see Euclidean group). Symmetry with respect to all rotations...

form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators...

of symmetries, since it is the polyhedron that is dual to an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or...

equivalently, symmetries on the sphere) with the largest symmetry groups. Icosahedral symmetry is not compatible with translational symmetry, so there are...

a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations)...

orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The group of all (not necessarily...

Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However...

The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes...

Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an...

one of them. The symmetry group of an object is sometimes also called its full symmetry group, as opposed to its proper symmetry group, the intersection...

dihedral symmetry (a 90° rotational symmetry, which also includes a symmetry on both orthogonal axis, 180° rotational symmetry, and diagonal symmetry) is known...

the set of plane symmetries that preserve the pattern forms a group. The groups that arise in this way are plane symmetry groups and are of considerable...

rotation) then this symmetry group is isomorphic to Z/nZ. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but...

according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group. Frieze groups are two-dimensional line groups, having repetition...

Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen...

In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional...