**Formal Treatment**

Formally, rotational symmetry is symmetry with respect to some or all rotations in *m*-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of *E*+(*m*) (see Euclidean group).

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole *E*(*m*). With the modified notion of symmetry for vector fields the symmetry group can also be *E*+(*m*).

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(*m*), the group of *m*×*m* orthogonal matrices with determinant 1. For this is the rotation group SO(3).

In another meaning of the word, the rotation group *of an object* is the symmetry group within *E*+(*n*), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also Rotational invariance.

Read more about this topic: Rotational Symmetry

### Other articles related to "formal treatment":

**Formal Treatment**

... Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition. ...

**Formal Treatment**- The Carlqvist Relation

... The Carlqvist Relation can be illustrated (see right), showing the total current (I) versus the number of particles per unit length (N) in a Bennett pinch ... The chart illustrates four physically distinct regions ...

### Famous quotes containing the words treatment and/or formal:

“To me, nothing can be more important than giving children books, It’s better to be giving books to children than drug *treatment* to them when they’re 15 years old. Did it ever occur to anyone that if you put nice libraries in public schools you wouldn’t have to put them in prisons?”

—Fran Lebowitz (20th century)

“It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a *formal* correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.”

—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)