**Three-dimensional and Four-dimensional Rotation Groups**

The term "conjugation", besides the meaning given above, can also mean taking an element *a* to *r a r*-1 where *r* is some non-zero element (quaternion). All elements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)

Thus the multiplicative group of non-zero quaternions acts by conjugation on the copy of **R**³ consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(*θ*) is a rotation by an angle 2*θ*, the axis of the rotation being the direction of the imaginary part. The advantages of quaternions are:

- Nonsingular representation (compared with Euler angles for example).
- More compact (and faster) than matrices.
- Pairs of unit quaternions represent a rotation in 4D space (see
*Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations*).

The set of all unit quaternions (versors) forms a 3-dimensional sphere *S*³ and a group (a Lie group) under multiplication, double covering the group *SO*(3,**R**) of real orthogonal 3×3 matrices of determinant 1 since *two* unit quaternions correspond to every rotation under the above correspondence.

The image of a subgroup of versors is a point group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefix **binary**. For instance, the preimage of the icosahedral group is the binary icosahedral group.

The versors' group is isomorphic to *SU*(2), the group of complex unitary 2×2 matrices of determinant 1.

Let *A* be the set of quaternions of the form *a* + *bi* + *cj* + *dk* where *a*, *b*, *c*, and *d* are either all integers or all rational numbers with odd numerator and denominator 2. The set *A* is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of a 24-cell regular polytope with Schläfli symbol {3,4,3}.

Read more about this topic: Quaternion

### Other articles related to "group, rotation, rotations":

**Three-dimensional and Four-dimensional Rotation Groups**

... Thus the multiplicative

**group**of non-zero quaternions acts by conjugation on the copy of R³ consisting of quaternions with real part equal to zero ... Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(θ) is a

**rotation**by an angle 2θ, the axis of the

**rotation**being the direction of the ... Pairs of unit quaternions represent a

**rotation**in 4D space (see

**Rotations**in 4-dimensional Euclidean space Algebra of 4D

**rotations**) ...

### Famous quotes containing the words groups and/or rotation:

“Trees appeared in *groups* and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”

—Vladimir Nabokov (1899–1977)

“The lazy manage to keep up with the earth’s *rotation* just as well as the industrious.”

—Mason Cooley (b. 1927)