Quaternion

{| class="wikitable" align="right" style="text-align:center; margin-left:0.5em; max-width: 230px;"

|+ Quaternion multiplication table

|width=15|

!width=15|

|colspan=5| Left column shows the left factor, top row shows the right factor. Also, <math>a\mathbf{b}=\mathbf{b}a</math> and <math>-\mathbf{b} = (-1)\mathbf{b}</math> for <math>a\in \mathbb{R} </math>, <math>\mathbf{b} = \mathbf{i}, \mathbf{j}, \mathbf{k} </math>.

thumb|[[Cayley graph of the quaternion group showing the six cycles of multiplication by , and . (If the image is opened in the Wikimedia Commons by clicking twice on it, cycles can be highlighted by hovering over or clicking on them.)]]

In mathematics, the quaternions form a number system similar to the complex numbers, with the usual arithmetical operations of addition, subtraction, multiplication, and division, but with four real-number components instead of two. Unlike with the complex numbers, quaternion multiplication is not commutative, meaning that the result of multiplying two quaternions depends on their order. Quaternions can be used to represent vectors in three-dimensional space, which provides a definition of the quotient of two vectors.

Quaternions were first described by the Irish mathematician and physicist William Rowan Hamilton in 1843, and in his honor the set of all quaternions is conventionally denoted by <math>\mathbb H</math> or . A generic quaternion is usually represented in the form

a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k,

</math>

where the coefficients , , , are real numbers, and , are the basis vectors or basis elements.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance imaging and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

As an abstract mathematical structure, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. Because of their non-commutative multiplication, they do not form a field. The quaternions are also a special case of a Clifford algebra, classified as <math>\operatorname{Cl}_{0,2}(\mathbb R)\cong \operatorname{Cl}_{3,0}^+(\mathbb R).</math>

According to the Frobenius theorem, the algebra <math>\mathbb H</math> is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra.

The unit quaternions give a group structure on the 3-sphere isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.

[[File:Quaternion 2.svg|thumb|right|Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually .

]]

History

right|thumb|Quaternion plaque on [[Broom Bridge|Brougham (Broom) Bridge, Dublin, which reads:

]]

Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.

Hamilton also called vector quaternions right quaternions and real numbers (considered as quaternions with zero vector part) scalar quaternions.

If a quaternion is divided up into its scalar part and its vector part, that is,

\mathbf q = (r,\,\vec{v}),\ \mathbf q \in \mathbb{H},\ r \in \mathbb{R},\ \vec{v}\in \mathbb{R}^3,

</math>

then the formulas for addition, multiplication, and multiplicative inverse are

<math display=block>\begin{align}

(r_1,\,\vec{v}_1) + (r_2,\,\vec{v}_2)

&= (r_1 + r_2,\,\vec{v}_1 + \vec{v}_2), \\[5mu]

(r_1,\,\vec{v}_1) (r_2,\,\vec{v}_2)

&= (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\,r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2), \\[5mu]

(r,\,\vec{v})^{-1}

&= \left(\frac{r}{r^2 + \vec{v}\cdot\vec{v,\ \frac{-\vec{v{r^2 + \vec{v}\cdot\vec{v\right)

\end{align}</math>

where "<math>{}\cdot{}</math>" and "<math>\times</math>" denote respectively the dot product and the cross product.

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let <math>q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k </math> be a quaternion. The conjugate of is the quaternion <math> q^* = a - b\,\mathbf i - c\,\mathbf j - d\,\mathbf k </math>. It is denoted by , qt, <math>\tilde q</math>, or . The real group ring of is a ring <math>\mathbb R[\mathrm Q_8]</math> which is also an eight-dimensional vector space over <math>\mathbb R.</math> It has one basis vector for each element of <math>\mathrm Q_8.</math> The quaternions are isomorphic to the quotient ring of <math>\mathbb R[\mathrm Q_8]</math> by the ideal generated by the elements , , , and . Here the first term in each of the sums is one of the basis elements , and , and the second term is one of basis elements , and , not the additive inverses of , and .

Quaternions and three-dimensional geometry

The vector part of a quaternion can be interpreted as a coordinate vector in <math>\mathbb R^3;</math> therefore, the algebraic operations of the quaternions reflect the geometry of <math>\mathbb R^3.</math> Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics. basis vectors of <math>\mathbb H</math> and a basis for <math>\mathbb R^3.</math> Replacing by , by , and by sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called the spatial inverse.

For two vector quaternions and their dot product, by analogy to vectors in <math>\mathbb R^3,</math> is

It can also be expressed in a component-free manner as

<math display=block>p \cdot q = \textstyle\frac{1}{2}(p^*q + q^*p) = \textstyle\frac{1}{2}(pq^* + qp^*).</math>

This is equal to the scalar parts of the products . Note that their vector parts are different.

The cross product of and relative to the orientation determined by the ordered basis , and is

<math display="block">p \times q = (c_1 d_2 - d_1 c_2)\mathbf i + (d_1 b_2 - b_1 d_2)\mathbf j + (b_1 c_2 - c_1 b_2)\mathbf k.</math>

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product (as quaternions), as well as the vector part of . It also has the formula

<math display=block>p \times q = \textstyle\tfrac{1}{2}(pq - qp).</math>

For the commutator, , of two vector quaternions one obtains

<math display=block>[p,q]= 2p \times q,</math>

which gives the commutation relationship

<math display=block>qp= pq - 2p \times q.</math>

In general, let and be quaternions and write

<math display=block>\begin{align}

p &= p_\text{s} + p_\text{v}, \\[5mu]

q &= q_\text{s} + q_\text{v},

\end{align}</math>

where and are the scalar parts, and and are the vector parts of and . Then we have the formula

<math display="block">pq = (pq)_\text{s} + (pq)_\text{v} = (p_\text{s}q_\text{s} - p_\text{v}\cdot q_\text{v}) + (p_\text{s} q_\text{v} + q_\text{s} p_\text{v} + p_\text{v} \times q_\text{v}).</math>

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points in Elliptic geometry.

Unit quaternions can be identified with rotations in <math>\mathbb R^3</math> and were called versors by Hamilton. for visualization of quaternions.

Matrix representations

Just as complex numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4 real matrices. In each case, the representation given is one of a family of linearly related representations. These are injective homomorphisms from <math>\mathbb H</math> to the matrix rings and , respectively.

Representation as complex 2 × 2 matrices

The quaternion can be represented using a complex as

<math display="block">\begin{bmatrix}

\phantom-a + bi & c + di \\

-c + di & a - bi

\end{bmatrix}.</math>

This representation has the following properties:

Constraining any two of , , and to zero produces a representation of complex numbers. For example, setting produces a diagonal complex matrix representation of complex numbers, and setting produces a real matrix representation.
The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
The scalar part of a quaternion is one half of the matrix trace.
The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
By restriction this representation yields a group isomorphism between the subgroup of unit quaternions and their image SU(2). Topologically, the unit quaternions are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group is important for describing spin in quantum mechanics; see Pauli matrices.
There is a strong relation between quaternions and Pauli matrices. The matrix above can be written as <math>a\,I + b\,i\,\sigma_3 + c\,i\,\sigma_2 + d\,i\,\sigma_1,</math> so in this representation the quaternion units correspond to <math>\left\{I,i\,\sigma_3,i\,\sigma_2,i\,\sigma_1\right\}</math> = <math>\left\{I,\sigma_1\,\sigma_2,\sigma_3\,\sigma_1,\sigma_2\,\sigma_3\right\}</math>. Multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for One obtains −1 via e.g. the last equality is <math display=block>\mathbf{i\;j\;k} = \sigma_1\,\sigma_2\,\sigma_3\,\sigma_1\,\sigma_2\,\sigma_3 = -1.</math>

The representation in is not unique: A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose <math display="block">

1 \mapsto \mathbf{I}, \quad \mathbf{i} \mapsto - i\,\sigma_1 = - \sigma_2\,\sigma_3, \quad \mathbf{j} \mapsto - i\,\sigma_2 = - \sigma_3\,\sigma_1, \quad \mathbf{k} \mapsto - i\,\sigma_3 = - \sigma_1\,\sigma_2,

</math>This gives an alternative representation,

a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k

\mapsto \begin{bmatrix}

a - d\,i & -c - b\,i \\

c - b\,i & \phantom-a + d\,i \end{bmatrix}.

</math>

Representation as real 4 × 4 matrices

Using 4 × 4 real matrices, that same quaternion can be written as

<math display=block>\begin{align}

\left[ \begin{array}{rrrr}

a & -b & -c & -d \\

b & a & -d & c \\

c & d & a & -b \\

d & -c & b & a

\end{array} \right]

&= a

\left[ \begin{array}{rrrr}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{array} \right]

+ b

\left[ \begin{array}{rrrr}

0 & -1 & 0 & 0 \\

1 & 0 & 0 & 0 \\

0 & 0 & 0 & -1 \\

0 & 0 & 1 & 0

\end{array} \right] \\[10mu]

&\qquad + c

\left[ \begin{array}{rrrr}

0 & 0 & -1 & 0 \\

0 & 0 & 0 & 1 \\

1 & 0 & 0 & 0 \\

0 & -1 & 0 & 0

\end{array} \right]

+ d

\left[ \begin{array}{rrrr}

0 & 0 & 0 & -1 \\

0 & 0 & -1 & 0 \\

0 & 1 & 0 & 0 \\

1 & 0 & 0 & 0

\end{array} \right].

\end{align}</math>

However, the representation of quaternions in is not unique. For example, the same quaternion can also be represented as

<math display=block>\begin{align}

\left[ \begin{array}{rrrr}

a & d & -b & -c \\

-d & a & c & -b \\

b & -c & a & -d \\

c & b & d & a

\end{array} \right]

&= a

\left[ \begin{array}{rrrr}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1

\end{array} \right]

+ b

\left[ \begin{array}{rrrr}

0 & 0 & -1 & 0 \\

0 & 0 & 0 & -1 \\

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0

\end{array} \right] \\[10mu]

&\qquad + c

\left[ \begin{array}{rrrr}

0 & 0 & 0 & -1 \\

0 & 0 & 1 & 0 \\

0 & -1 & 0 & 0 \\

1 & 0 & 0 & 0

\end{array} \right]

+ d

\left[ \begin{array}{rrrr}

0 & 1 & 0 & 0 \\

-1 & 0 & 0 & 0 \\

0 & 0 & 0 & -1 \\

0 & 0 & 1 & 0

\end{array} \right].

\end{align}</math>

There are 48 distinct matrix representations of this form, in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48 sets of quadruples of matrices with these symmetry constraints, such that a function sending , and to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.

In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two by setting

Each representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table

{|class="wikitable" style="text-align:center"

|width=15|

!width=15|

which is isomorphic, through <math>\{a \mapsto 1,\ b \mapsto i,\ c \mapsto j,\ d \mapsto k\},</math> to

{|class="wikitable" style="text-align:center"

|width=15|

!width=15|

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing with , with , with , and with and removing the row and column headers yields a matrix representation of .

Lagrange's four-square theorem

Quaternions are also used in one of the proofs of Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.

Quaternions as pairs of complex numbers

Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

Let <math>\mathbb C^2</math> be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements and . A vector in <math>\C^2</math> can be written in terms of the basis elements and as

<math display=block>(a + b\,i) 1 + (c + d\,i) \mathbf j .</math>

If we define and , then we can multiply two vectors using the distributive law. Using as an abbreviated notation for the product leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion . If we write the elements of <math>\mathbb C^2</math> as ordered pairs and quaternions as quadruples, then the correspondence is

<math display=block>(a + bi,\,c + di) \leftrightarrow (a,\,b,\,c,\,d).</math>

Square roots

Square roots of −1

In the complex numbers, <math>\C,</math> there are exactly two numbers, and , that give −1 when squared. In <math>\mathbb H</math> there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in <math>\R^3.</math> To see this, let be a quaternion, and assume that its square is −1. In terms of , , , and , this means

<math display=block>\begin{align}

a^2 - b^2 - c^2 - d^2 &= -1, \vphantom{x^|} \\[3mu]

2ab &= 0, \\[3mu]

2ac &= 0, \\[3mu]

2ad &= 0.

\end{align}</math>

To satisfy the last three equations, either or , , and are all The latter is impossible because is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to if and only if it is a vector quaternion with By definition, the set of all such vectors forms the unit sphere.

Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0).

As a union of complex planes

Each antipodal pair of square roots of creates a distinct copy of the complex numbers inside the quaternions. If then the copy is the image of the function

<math display=block>a + bi \mapsto a + b q.</math>

This is an injective ring homomorphism from <math>\mathbb C</math> to <math>\mathbb H,</math> which defines a field isomorphism from <math>\Complex</math> onto its image. The images of the embeddings corresponding to and are identical.

Every non-real quaternion generates a subalgebra of the quaternions that is isomorphic to <math>\C</math>, and is thus a planar subspace of <math>\mathbb H</math>: write as the sum of its scalar part and its vector part:

q = q_s + \vec{q}_v.

</math>

Decompose the vector part further as the product of its norm and its versor:

q = q_s + \lVert \vec{q}_v \rVert \cdot \mathbf{U}\,\vec{q}_v = q_s + \|\vec q_v\|\,\frac{ \vec q_v }{ \|\vec q_v\| } .

</math>

(This is not the same as <math>q_s + \lVert q\rVert\cdot\mathbf{U}q .</math>) The versor of the vector part of , <math>\mathbf{U}\vec{q}_v,</math> is a right versor with –1 as its square. A straightforward verification shows that

<math display=block>a + bi \mapsto a + b\mathbf{U}\vec{q}_v</math>

defines an injective homomorphism of normed algebras from <math>\mathbb C</math> into the quaternions. Under this homomorphism, is the image of the complex number <math>q_s + \lVert\vec{q}_v\rVert i .</math>

As <math>\mathbb H</math> is the union of the images of all these homomorphisms, one can view the quaternions as a pencil of planes intersecting on the real line. Each of these complex planes contains exactly one pair of antipodal points of the sphere of square roots of minus one.

Commutative subrings

The relationship of quaternions to each other within the complex subplanes of <math>\mathbb H</math> can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions and commute (i.e., ) only if they lie in the same complex subplane of <math>\mathbb H,</math> the profile of <math>\mathbb H</math> as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring.

Square roots of arbitrary quaternions

Any quaternion <math>\mathbf q = (r,\, \vec{v})</math> (represented here in scalar–vector representation) has at least one square root <math>\sqrt{\mathbf q} = (x,\, \vec{y})</math> which solves the equation <math>(\sqrt{\mathbf q})^{2} = (x,\,\vec{y})^2 = \mathbf q .</math> Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions

\sqrt{\mathbf q}

= \sqrt{(r,\, \vec{v})}

= \pm \left( \sqrt{ \tfrac{1}{2} \bigl( \|\mathbf q \| + r \bigr)},\

\frac{ \vec{v} }{ \|\vec{v}\| }\sqrt{ \tfrac{1}{2} \bigl( \|\mathbf q\| - r \bigr)}\right),

</math>

where <math display="inline">\|\vec{v}\| = \sqrt{\vec{v}\cdot\vec{v</math> is the norm of <math>\vec{v}</math> and <math display="inline">\|\mathbf q\| = \sqrt{ \mathbf q^*\mathbf q} = \sqrt{ r^2 + \|\vec{v}\|^2}</math> is the norm of <math>\mathbf q .</math> For any scalar quaternion <math>\mathbf q</math>, this equation provides the correct square roots if <math display="inline">\vec{v} / \|\vec{v}\|</math> is interpreted as an arbitrary unit vector.

Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on <math>\{ 0 \} \times S^2 \bigl( \sqrt{-r} \bigr)</math>, i.e., where the scalar part is zero and the vector part is located on the 2-sphere with radius <math>\sqrt{-r} .</math>

Functions of a quaternion variable

thumb|The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3 dimensions. This Julia set is cross sectioned at the plane.

Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of the Mandelbrot set and Julia sets into 4-dimensional space.

Exponential, logarithm, and power functions

A function of a quaternion can be defined from a power series with real coefficients. For example, given a quaternion,

<math display=block>q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k = a + \mathbf{v},</math>

the exponential is computed as

\exp(q) = \sum_{n=0}^\infty \frac{q^n}{n!}

= e^{a} \biggl({\cos \|\mathbf{v}\|} + \frac{ \mathbf{v} }{ \|\mathbf{v}\| } \sin \|\mathbf{v}\| \biggr),

</math>

and the logarithm is often use different notation with — that is, another variable .

<math display=block>a = \| q \|\, \cos( \varphi )</math>

and the unit vector <math>\hat{n}</math> is defined by:

<math display=block>\mathbf{v} = \hat{n} \|\mathbf{v}\| = \hat{n}\|q\|\,\sin(\varphi).</math>

Any unit quaternion may be expressed in polar form as:

<math display=block>q = \exp{(\hat{n}\varphi)}.</math>

The power of a quaternion raised to an arbitrary (real) exponent is given by:

<math display=block>q^x = \|q\|^x e^{\hat{n} x \varphi} = \|q\|^x \bigl(\cos(x\varphi) + \hat{n} \sin(x\varphi) \bigr).</math>

Geodesic norm

The geodesic distance between unit quaternions and is defined as:

<math display=block>d_\text{g}(p, q) = \lVert \ln(p^{-1} q) \rVert .</math>

and amounts to the absolute value of half the angle subtended by and along a great arc of the sphere.

This angle can also be computed from the quaternion dot product without the logarithm as:

<math display=block>d_\text{g}(p, q) = {\arccos}\bigl(2(p \cdot q)^2 - 1 \bigr).</math>

Three-dimensional and four-dimensional rotation groups

The word "conjugation", besides the meaning given above, can also mean taking an element to where is some nonzero 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 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 is a rotation by an angle , the axis of the rotation being the direction of the vector part. The advantages of quaternions are:

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 4D space (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations).

The set of all unit quaternions (versors) forms a 3-sphere and a group (a Lie group) under multiplication, double covering the group <math>\text{SO}(3,\mathbb{R})</math> of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See plate trick.

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 , the group of complex unitary of determinant 1.

Let be the set of quaternions of the form where and are either all integers or all half-integers. The set 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 regular 24 cell with Schläfli symbol They correspond to the double cover of the rotational symmetry group of the regular tetrahedron. Similarly, the vertices of a regular 600 cell with Schläfli symbol can be taken as the unit icosians, 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.

Quaternion algebras

The Quaternions can be generalized into further algebras called quaternion algebras. Take to be any field with characteristic different from 2, and and to be elements of ; a four-dimensional unitary associative algebra can be defined over with basis and , where , and (so ).

Quaternion algebras are isomorphic to the algebra of 2×2 matrices over or form division algebras over , depending on the choice of and .

Quaternions as the even part of

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}^+ (\R)</math> of the Clifford algebra <math>\operatorname{Cl}_{3,0}(\R).</math> This is an associative multivector algebra built up from fundamental basis elements 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 in a plane perpendicular to a unit vector 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 σ1 and σ2. 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 bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directions and , there is only one bivector basis element , so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements , , , 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 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:

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 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 spaces, for example, the Minkowski space of special relativity. In such spaces rotors can be used to efficiently represent Lorentz boosts, and to interpret formulas involving the gamma matrices.

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

Brauer group

The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers is Brauer equivalent to either the real numbers or the quaternions. Explicitly, the Brauer group of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.

CSAs – finite dimensional rings over a field, which are simple algebras (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of extension fields, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.

Notes

References

External links

Paulson, Lawrence C. Quaternions (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
Quaternions – Visualisation

History

Conjugation, the norm, and reciprocal

Quaternions and three-dimensional geometry

Matrix representations

Representation as complex 2 × 2 matrices

Representation as real 4 × 4 matrices

Lagrange's four-square theorem

Quaternions as pairs of complex numbers

Square roots

Square roots of −1

As a union of complex planes

Commutative subrings

Square roots of arbitrary quaternions

Functions of a quaternion variable

Exponential, logarithm, and power functions

Geodesic norm

Three-dimensional and four-dimensional rotation groups

Quaternion algebras

Quaternions as the even part of

Brauer group

See also

Notes

References

Further reading

Books and publications

Links and monographs

External links