Three-dimensional and four-dimensional rotation groups

The appellation "conjugation", besides the acceptation accustomed above, can aswell beggarly demography an aspect a to r a r-1 area r is some non-zero aspect (quaternion). All elements that are conjugate to a accustomed aspect (in this faculty of the chat conjugate) accept the aforementioned complete allotment and the aforementioned barometer of the agent part. (Thus the conjugate in the added faculty is one of the conjugates in this sense.)

Thus the multiplicative accumulation of non-zero quaternions acts by alliance on the archetype of R³ consisting of quaternions with complete allotment according to zero. Alliance by a assemblage quaternion (a quaternion of complete amount 1) with complete allotment cos(θ) is a circling by an bend 2θ, the arbor of the circling getting the administration of the abstract part. The advantages of quaternions are:

Non atypical representation (compared with Euler angles for example).

More bunched (and faster) than matrices.

Pairs of assemblage quaternions represent a circling in 4D amplitude (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations).

The set of all assemblage quaternions (versors) forms a 3-dimensional apple S³ and a accumulation (a Lie group) beneath multiplication, bifold accoutrement the accumulation SO(3,R) of complete erect 3×3 matrices of account 1 back two assemblage quaternions accord to every circling beneath the aloft correspondence.

For added data on this topic, see Point groups in three dimensions.

The angel of a subgroup of versors is a point group, and conversely, the preimage of a point accumulation is a subgroup of versors. The preimage of a bound point accumulation is alleged by the aforementioned name, with the prefix binary. For instance, the preimage of the icosahedral accumulation is the bifold icosahedral group.

The versors' accumulation is isomorphic to SU(2), the accumulation of circuitous unitary 2×2 matrices of account 1.

Let A be the set of quaternions of the anatomy a + bi + cj + dk area 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 actuality a domain) and a filigree and is alleged the ring of Hurwitz quaternions. There are 24 assemblage quaternions in this ring, and they are the vertices of a 24-cell approved polytope with Schläfli attribute {3,4,3}.