Editing Quaternion (section)

== Quaternions as the even part of {{math|Cl<sub>3,0</sub>(R)}} ==
{{main|Spinor#Three dimensions}}
The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even part <math>\operatorname{Cl}_{3,0}^+(\mathbb R)</math> of the Clifford algebra <math>\operatorname{Cl}_{3,0}(\mathbb R).</math>  This is an associative multivector algebra built up from fundamental basis elements {{math|[[Pauli matrices|''σ''<sub>1</sub>, ''σ''<sub>2</sub>, ''σ''<sub>3</sub>]]}} using the product rules

<math display=block>\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 1,</math>
<math display=block>\sigma_m \sigma_n = - \sigma_n \sigma_m \qquad (m \neq n).</math>

If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that the ''reflection'' of a vector {{mvar|r}} in a plane perpendicular to a unit vector {{mvar|w}} can be written:

<math display=block>r^{\prime} = - w\, r\, w.</math>

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

<math display=block>r^{\prime\prime} = \sigma_2 \sigma_1 \, r \, \sigma_1 \sigma_2 </math>

corresponds to a rotation of 180° in the plane containing ''σ''<sub>1</sub> and ''σ''<sub>2</sub>.  This is very similar to the corresponding quaternion formula,

<math display=block>r^{\prime\prime} = -\mathbf{k}\, r\, \mathbf{k}. </math>

Indeed, the two structures <math>\operatorname{Cl}_{3,0}^+(\mathbb R)</math> and <math>\mathbb H</math> are [[isomorphic]]. One natural identification is

<math display=block>1 \mapsto 1, \quad \mathbf{i} \mapsto - \sigma_2 \sigma_3, \quad \mathbf{j} \mapsto - \sigma_3 \sigma_1, \quad \mathbf{k} \mapsto - \sigma_1 \sigma_2,</math>

and it is straightforward to confirm that this preserves the Hamilton relations

<math display=block>\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i \,j \,k} = -1.</math>

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to [[bivector]]s – quantities with [[magnitude (mathematics)|magnitudes]] and [[orientation (mathematics)|orientations]] associated with particular 2D&nbsp;''planes'' rather than 1D&nbsp;''directions''. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions {{math|''σ''<sub>1</sub>}} and {{math|''σ''<sub>2</sub>}}, there is only one bivector basis element {{math|''σ''<sub>1</sub>''σ''<sub>2</sub>}}, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements {{math|''σ''<sub>2</sub>''σ''<sub>3</sub>}}, {{math|''σ''<sub>3</sub>''σ''<sub>1</sub>}}, {{math|''σ''<sub>1</sub>''σ''<sub>2</sub>}}, so three imaginaries.

This reasoning extends further. In the Clifford algebra <math>\operatorname{Cl}_{4,0}(\mathbb R),</math> there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, called [[rotor (mathematics)|rotors]], can be very useful for applications involving [[homogeneous coordinates]].  But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as a [[pseudovector]].

There are several advantages for placing quaternions in this wider setting:<ref>{{cite web |url=http://www.geometricalgebra.net/quaternions.html |title=Quaternions and Geometric Algebra |website=geometricalgebra.net |access-date=2008-09-12}} See also: {{cite book |first1=Leo |last1=Dorst |first2=Daniel |last2=Fontijne |first3=Stephen |last3=Mann |year=2007 |url=http://www.geometricalgebra.net/index.html |title=Geometric Algebra for Computer Science |publisher=[[Morgan Kaufmann]] |isbn=978-0-12-369465-2}}</ref>
* Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
* In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
* Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
* In the [[Conformal geometric algebra|conformal model]] of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
* Rotor-encoded transformations make interpolation particularly straightforward.
* Rotors carry over naturally to [[pseudo-Euclidean space]]s, for example, the [[Minkowski space]] of [[special relativity]]. In such spaces rotors can be used to efficiently represent [[Lorentz boost]]s, and to interpret formulas involving the [[gamma matrices]].{{cn|date=October 2024}}

For further detail about the geometrical uses of Clifford algebras, see [[Geometric algebra]].