Editing Quaternion (section)

==Three-dimensional and four-dimensional rotation groups==
{{Main|Quaternions and spatial rotation|Rotation operator (vector space)}}
The word "[[conjugation (group theory)|conjugation]]", besides the meaning given above, can also mean taking an element {{mvar|a}} to {{math|''r a r''<sup>−1</sup>}} where {{mvar|r}} is some nonzero quaternion. All [[conjugacy class|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.) <ref>{{cite arXiv |last=Hanson |first=Jason |year=2011 |title=Rotations in three, four, and five dimensions |class=math.MG |eprint=1103.5263 }}</ref>

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of <math>\mathbb R^3</math> consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part {{math|cos(''φ'')}} is a rotation by an angle {{math|2''φ''}}, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:<ref>{{cite thesis |type=BS |last=Günaşti |first=Gökmen |year=2016 |title=Quaternions Algebra, Their Applications in Rotations and Beyond Quaternions |publisher=Linnaeus University |url=http://www.diva-portal.org/smash/get/diva2:535712/FULLTEXT02 }}</ref>

* Avoiding [[gimbal lock]], a problem with systems such as Euler angles.
* Faster and more compact than matrices.
* Nonsingular representation (compared with Euler angles for example).
* Pairs of unit quaternions represent a rotation in [[Four-dimensional space|4D]] space (see ''[[Rotations in 4-dimensional Euclidean space#Algebra of 4D rotations|Rotations in 4-dimensional Euclidean space: Algebra of 4D&nbsp;rotations]]'').

<!-- ''S''³ itself has not a canonical group structure -->The set of all unit quaternions ([[versor]]s) forms a 3-sphere {{math|''S''<sup>3</sup>}} and a group (a [[Lie group]]) under multiplication, [[Covering space#Properties|double covering]] the group <math>\text{SO}(3,\mathbb{R})</math> of real orthogonal 3×3&nbsp;[[orthogonal matrix|matrices]] of [[determinant]]&nbsp;1 since ''two'' unit quaternions correspond to every rotation under the above correspondence. See [[plate trick]].
{{further|Point groups in three dimensions}}
The image of a subgroup of versors  is a [[Point groups in three dimensions|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 {{math|SU(2)}}, the group of complex [[unitary matrix|unitary]] 2×2&nbsp;matrices of [[determinant]] 1.

Let {{mvar|A}} be the set of quaternions of the form {{nowrap|{{math|''a'' + ''b'' '''i''' + ''c'' '''j''' + ''d'' '''k'''}} }} where {{mvar|a, b, c,}} and {{mvar|d}} are either all [[integer]]s or all [[half-integer]]s. The set {{mvar|A}} is a ring (in fact a [[domain (ring theory)|domain]]) and a [[Lattice (group)|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 24 cell]] with [[Schläfli symbol]] {{math|{3,4,3}.}} They correspond to the double cover of the rotational symmetry group of the regular [[tetrahedron]]. Similarly, the vertices of a [[600-cell|regular 600 cell]] with Schläfli symbol {{math|{3,3,5}}} can be taken as the unit [[icosian]]s, corresponding to the double cover of the rotational symmetry group of the [[regular icosahedron]]. The double cover of the rotational symmetry group of the regular [[octahedron]] corresponds to the quaternions that represent the vertices of the [[disphenoidal 288-cell]].<ref>{{Cite web |title=Three-Dimensional Point Groups |url=https://www.classe.cornell.edu/~dms79/xrd/xtallography/Three-Dimensional%20Point%20Groups.htm |access-date=2022-12-09 |website=www.classe.cornell.edu}}</ref>