Quaternions

{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;" |+ Quaternion multiplication table |- |width=15| !width=15| !width=15| !width=15| !width=15| |- ! | | | | |- ! | | | | |- ! | | | | |- ! | | | | |- |colspan=5| Left column shows the left factor, top row shows the right factor. Also, a\mathbf{b}=\mathbf{b}a and -\mathbf{b} = (-1)\mathbf{b} for a\in \mathbb{R} , \mathbf{b} = \mathbf{i}, \mathbf{j}, \mathbf{k} . |} thumb|Cayley graph of the [[quaternion group showing the six cycles of multiplication by , and . (If the image

geometric space with four dimensions

product of sums of four squares is a sum of four squares

finite group with 8 elements, whose elements can be represented by multiplication of unit quaternions {±1, ±i, ±j, ±k}

thumb|Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the next picture the 3D space contained the shadow of the bulk hypersphere. thumb|Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spher

In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.

theorem that the finite-dimensional associative division algebras over the reals are either the reals, the complex numbers, or the quaternions

type of cyclic group in group theory

correspondence between quaternions and 3D rotations

In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form \ u = \exp(a\mathbf{r}) = \cos a + \mathbf{r} \sin a, \qquad \mathbf{r}^2 = -1, \qquad a \in [0,\pi]\ , where the condition \ \mathbf{r}^2 = -1\ means that \ \mathbf{r}\ is an algebraic imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit \ \mathbf{r} ~. If \ a = \tfrac{\pi}{2}\ (when \ a\ is a right angle), then \ u = \mathbf{r}\ , and it is called a right versor.

special orthogonal group

{|class="wikitable" align="right" style="text-align:center" |+Split-quaternion multiplication |- !width=15| × !width=15| 1 !width=15| i !width=15| j !width=15| k |- ! 1 | 1 | i | j | k |- !i |i |−1 |k |−j |- !j |j |−k |1 |−i |- !k |k |j |i |1 |}

a quaternion whose components are either all integers or all half-integers

generalization of quaternions to other fields

method of quaternion rotation

eight-dimensional algebra over the real numbers

Riemannian manifold with Sp(n) holonomy, or equivalently with an S² family of Kähler structures