quaternion n : the cardinal number that is the sum of three and one [syn: four, 4, IV, tetrad, quatern, quaternary, quaternity, quartet, quadruplet, foursome, Little Joe]

English

Etymology

From quaternioun, from quaterniō, from quaternī (ablative form of quater, “four times”) + -iōn, “-ion”.

Pronunciation

• a RP /kwɒˈtɜːniɒn/

Noun

1. A group or set of four people or things.
2. In the context of "mathematics": A four-dimensional hypercomplex number that consists of a real dimension and 3 imaginary ones (i, j, k) that are each a square root of -1. They are commonly used in vector mathematics and three-dimensional games. Quaternion multiplication is the most common technique used by mathematicians to avoid gimbal lock when rotating vectors.

Translations

References

expert-subject Mathematics In mathematics, quaternions are a non-commutative extension of complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. At first, quaternions were regarded as pathological because they disobeyed the commutative law ab = ba. Although they have been superseded in most applications by vectors and matrices, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations, such as in 3D computer graphics.

In modern language, quaternions form a 4-dimensional normed division algebra over the real numbers. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by \mathbb (Unicode ℍ). It can also be given by the Clifford algebra classifications Cℓ0,2(R) = Cℓ03,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only three finite-dimensional division rings containing the real numbers as a subring.

Definition

The quaternions are defined as the ring:

\mathbb=\

where addition is defined by:

(a_1+b_1i+c_1j+d_1k)+(a_2+b_2i+c_2j+d_2k)\,

=(a_1+a_2)+(b_1+b_2)i+(c_1+c_2)j+(d_1+d_2)k\,

and multiplication is defined by expanding:

(a_1+b_1i+c_1j+d_1k)(a_2+b_2i+c_2j+d_2k)\,

using the distributive law and then applying the defining relations:

i^2 = j^2 = k^2 = ijk = -1,\,

Every quaternion is a unique and real linear combination of the basis quaternions 1, i, j, and k.

Properties

Basis multiplication

The set of equations

i^2 = j^2 = k^2 = i j k = -1 , \,\!

where i, j, and k are imaginary numbers, is the fundamental formula for quaternion multiplicative identities, summarized in the multiplication table of basis quaternions.

\begin

ij & = & k, & & & & ji & = & -k, \\ jk & = & i, & & & & kj & = & -i, \\ ki & = & j, & & & & ik & = & -j. \end

For example, since

- 1 = i j k, \,\!

right-multiplying both sides by k gives

\begin -k & = & i j k k, \\ -k & = & i j (-1), \\ k & = & i j. \end \,\!

The rest of the table can be verified similarly.

Unlike multiplication of real or complex numbers, multiplication of quaternions is not commutative: e.g. ij = k, while ji = -k. The non-commutativity of multiplication has some unexpected consequences, among them that polynomial equations over the quaternions can have more distinct solutions than the degree of the polynomial. The equation z^2 + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b^2 + c^2 + d^2 = 1, so that these solutions form a unitary sphere centered on zero in the three-dimensional pure imaginary subspace of quaternions, this imaginary sphere intersecting the complex plane only at the two poles i and -i.

Algebras

The set H of all quaternions is a vector space over the real numbers with dimension 4 (the complex numbers have dimension 2 by comparison). While H is a four-dimensional vector space, one speaks of the scalar part of the quaternion as being a, while the vector part is the remainder bi + cj + dk. Thus, in the context of quaternions, a quaternion with zero for its scalar part is a vector.

Addition of quaternions is accomplished by adding corresponding coefficients, as with the complex numbers. By linearity, multiplication of quaternions is completely determined by the multiplication table above for the basis quaternions. Under this multiplication, the basis quaternions, with their negatives, form the quaternion group of order 8, Q_8.

The quaternions are an example of a division ring, an algebraic structure similar to a field except for commutativity of multiplication. In particular, multiplication is still associative and every non-zero element has a unique multiplicative inverse.

Quaternions form a 4-dimensional associative algebra over the reals (in fact a division algebra) and contain the complex numbers, but they do not form an associative algebra over the complex numbers. The quaternions, along with the complex and real numbers, are the only finite-dimensional associative division algebras over the field of real numbers.

Quaternion operations

Quaternion operations have extended applications in electrodynamics, general relativity, and 3D graphics programming. The use of quaternions can replace tensors in representation. It is sometimes easier to use quaternions with complex elements, leading to a form that is not a division algebra. However, the same operations can be performed using a combination of conjugate operations. Only quaternions with real elements will be discussed here.

Definitions used in this section

This section, used to describe common algebraic operations on quaternions, will define three quaternions. These quaternions will be used to represent a primary operand, a secondary operand, and a resultant. These are respectively: A, B, and Q. Not all operations are complex enough to require their display using all three quaternions.

\begin\mathbf A & \equiv A_t & + & A_x & + & A_y & + & A_z\end

\begin\mathbf B & \equiv B_t & + & B_x & + & B_y & + & B_z\end

\begin\mathbf Q & \equiv Q_t & + & Q_x & + & Q_y & + & Q_z\end

Not all representations of quaternions may define the elements in the same way. These axes are chosen to, hopefully, aid in the description. The t element represents the scalar quantity. In this situation, the number 1 can be represented by the quaternion 1 + 0 + 0 + 0, such that the 1 would be in the t location.

The vector form of a quaternion may also be used. This form assumes that \vec \equiv A_x\mathbf i + A_y\mathbf j + A_z\mathbf k.

\equiv A_t + \vec A

\equiv B_t + \vec B

\equiv Q_t + \vec Q

Example cases will require that the defined quaternions above have example values:

let \begin\mathbf A & = & 3 & + & \mathbf i\end

let \begin\mathbf B & = & 5 \mathbf i & + & \mathbf j & - & 2 \mathbf k\end

Antiautomorphisms

The negation operation corresponds to the negation operation of the Clifford Algebras, in that the negation operation is mapped to all elements.

-\mathbf A \equiv -A_t - A_x \mathbf i - A_y\mathbf j - A_z\mathbf k

-\mathbf A \equiv -A_t - \vec A

The quaternion conjugate corresponds to the reversal operation of the Clifford algebras. The term Spatial inverse refers to the negation of each of the elements that would have a spatial representation, which are the elements in the i basis, the j basis, and the k basis.

NOTE: The operator symbol for the conjugate is not standardized. This can sometimes be seen as \overline\,\!, \tilde\,\!, Q^*\,\!, Q^t\,\!, and sometimes other symbols are used. Later in this article, \overline\,\! is used to denote the conjugate.

\overline \equiv A_t - A_x\mathbf i - A_y\mathbf j - A_z\mathbf k

\overline \equiv A_t - \vec

Common binary operations

Addition is the simple map of the addition operator over each element in the quaternions.

\mathbf A + \mathbf B \equiv (A_t + B_t) + (A_x + B_x)\mathbf i + (A_y + B_y)\mathbf j + (A_z + B_z)\mathbf k

\mathbf A + \mathbf B \equiv (A_t + B_t) + \vec A + \vec B

Again, subtraction is a map of the subtraction operator over each element. This is equivalent to using addition with the negation operations.

\mathbf A - \mathbf B \equiv (A_t - B_t) + (A_x - B_x)\mathbf i + (A_y - B_y)\mathbf j + (A_z - B_z)\mathbf k

\mathbf A - \mathbf B \equiv (A_t - B_t) + \vec A - \vec B

Quaternion products

The most useful quaternion product is the Grassmann product, which is non-commutative. There are times that the Grassmann product can be commutative and times that the Grassmann product can be anticommutative--this is because the first three operators are commutative and the cross product is anticommutative. The operation is usually denoted as the concatenation of one quaternion with another.

let \mathbf Q = \mathbf = A_t B_t - \vec\cdot\vec + A_t\vec + B_t\vec + \vec\times\vec

The components of Q:

\beginQ_t & = & A_t B_t & - & A_x B_x & - & A_y B_y & - & A_z B_z\end

\beginQ_x & = & A_t B_x & + & A_x B_t & + & A_y B_z & - & A_z B_y\end

\beginQ_y & = & A_t B_y & - & A_x B_z & + & A_y B_t & + & A_z B_x\end

\beginQ_z & = & A_t B_z & + & A_x B_y & - & A_y B_x & + & A_z B_t\end

It should be noted at this point that the anticommutative part of the product is the cross product of the vectors \left(\vec\times\vec\right). The remainder of the product is the commutative portion. If there is no anticommutative part to sum, then the product is entirely commutative. An example of a commutative product with a quaternion is any scalar value multiplied by a quaternion.

Properties:

• Non-commutative: for some \mathbf A and \mathbf B, \mathbf \neq \mathbf.
• Associative: \mathbf = \mathbf = \mathbf
• Left and Right Distributive: \mathbf = \mathbf,\quad \mathbf=\mathbf

The inner product (also called the quaternion dot-product) corresponds to the sum of the products of the individual elements. It is an entirely commutative product that returns a scalar quantity.

\mathbf A \cdot \mathbf B \equiv \mathbf B \cdot \mathbf A = A_t B_t + A_x B_x + A_y B_y + A_z B_z\,\!

Example:

\mathbf A \cdot \mathbf B = (3\cdot 0) + (1\cdot 5) + (0\cdot 1) + (0\cdot -2) = 5\,\!

In terms of the Grassmann product:

\mathbf A \cdot \mathbf B = \frac

This product is useful to isolate an element from a quaternion. For instance, the i term can be pulled out from p:

\mathbf A \cdot i = A_x \,\!

Properties:

• Commutative: \mathbf = \mathbf\,\!
• Associative: \mathbf = \mathbf = \mathbf\,\!
• Distributive: \mathbf = \mathbf + \mathbf

The outer-product is not used often; however, it is mentioned as a pair with the inner-product:

\operatorname(\mathbf A,\mathbf B) = A_t \vec - B_t\vec - \vec\times\vec\,\!

The outer-product can be rewritten using the Grassmann product:

\operatorname(\mathbf A, \mathbf B) = \frac \,\!

and the absolute value of p is the non-negative real number defined by

|p| = \sqrt = \sqrt. \,\!

where p^* := a - bi - cj - dk is the conjugate of p.

Note that (q p)* = p* q*, which is not in general equal to q* p*. The multiplicative inverse of a non-zero quaternion p can be conveniently computed as p−1 = p* / |p|².

By using the distance function d(p, q) = |p − q|, the quaternions form a metric space (isometric to the usual Euclidean metric on R4) and the arithmetic operations are continuous. We also have |p q| = |p| |q| for all quaternions p and q. Using the absolute value as norm, the quaternions form a real Banach algebra.

Given quarternions

p = a+\vec,\quad q = t+\vec,

with

\vec = bi + cj + dk,\quad\ \vec = xi + yj + zk, some other products are defined as follows.

The cross-product of quaternions is also known as the odd-product or the Grassmann outer-product. It is equivalent to the vector cross-product, and returns a vector quantity only:

p \times q = \vec\times\vec \,\!

p \times q = (cz - dy)i + (dx - bz)j + (by - cx)k \,\!

The cross-product can be rewritten using the Grassmann product:

p \times q = \frac \,\!

The even-product of quaternions is also referred to as the Grassmann inner-product. It is also not widely used, but it mentioned due to the similarity between it and the odd-product. It is the purely symmetric product; therefore, it is completely commutative.

\operatorname(p,q) = at - \vec\cdot\vec + a\vec + t\vec \,\!

\operatorname(p,q) = (at - bx - cy - dz) + (ax + bt)i + (ay + ct)j + (az + dt)k \,\!

The even-product can be rewritten using the Grassmann product:

\operatorname(p,q) = \frac \,\!

Another multiplication between two quaternions is termed the Euclidean product. Instead of the first quaternion, its conjugate is taken:

p^*q = at + \vec\cdot\vec + a\vec - t\vec - \vec\times\vec \,\!

Due to the non-commutative nature of the quaternion multiplication, p*q is not equivalent to q*p.

q^*p = at + \vec\cdot\vec - a\vec + t\vec + \vec\times\vec \,\!

When p = q, the result is the square of the absolute value.

The inverse of a quaternion is defined in a way that p−1p = pp−1 = 1. It is formed the same way that the complex inverse is found:

p^ = \frac \,\!

The inner product of a quaternion and its conjugate is a scalar. The division of a quaternion by a scalar is equivalent to multiplication by the scalar inverse, such that each element of the quaternion is divided by the divisor.

The non-commutativity of quaternions allows for two divisions of numbers p−1 q and q p−1. This means that the notation of q/p is ambiguous unless p is a scalar, q is a scalar, or an explicit convention is defined, which is not normally done.

The scalar of a quaternion can be isolated in the same way that was described earlier with the dot-product:

1\cdot p = \frac = a \,\!

The vector of a quaternion can be isolated using the outer-product in the same way the inner product is used to isolate the scalar:

\operatorname(1, p) = \frac = \vec = bi + cj + dk \,\!

The absolute value of a quaternion is the scalar quantity that determines the length of the quaternion from the origin.

|p| = \sqrt = \sqrt = \sqrt \,\!

The sign of a complex number finds the complex number of the same direction found on the unit circle. The unit quaternion is defined similarly as the quaternion in the same direction on the unit 4-dimensional hypersphere. The quaternion sign function produces the unit quaternion:

\sgn(p) = \frac \,\!

The argument finds the angle of the 4-vector quaternion from the unit scalar (i.e. 1). This returns a scalar angle.

\arg(p) = \arccos\left(\frac\right) \,\!

Example

Let

\begin

x & = & 3 + i \\ y & = & 5i + j - 2k \end Then

\begin

x + y & = & 3 + 6i + j - 2k \\ \\ xy & = & (3 + i)(5i + j - 2k) \\ & = & 15i + 3j - 6k + 5i^2 + ij - 2ik \\ & = & 15i + 3j - 6k - 5 + k + 2j \\ & = & -5 + 15i + 5j - 5k \\ \\ yx & = & (5i + j - 2k)(3 + i) \\ & = & 15i + 5i^2 + 3j + ji - 6k - 2ki \\ & = & 15i - 5 + 3j - k - 6k - 2j \\ & = & -5 + 15i + j - 7k \end

Matrix representations

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 (i.e., quaternion-matrix homomorphisms). One is to use 2×2 complex matrices, and the other is to use 4×4 real matrices.

Using 2×2 complex matrices, the quaternion a + b i + c j + d k can be represented as

\left(\begin a+bi & c+di \\ -c+di & a-bi \end\right)

This representation has the following properties:

• Complex numbers (c = d = 0) correspond to diagonal matrices.
• The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
• Restricted to unit quaternions, this representation provides an isomorphism between S3 and SU(2). The latter group is important in quantum mechanics when dealing with spin; see also Pauli matrices.

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

\left(\begin

a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b \\ -d & -c & b & a \end\right)

= a

\left(\begin 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end\right) + b \left(\begin 0 & \;\; 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & \;\; 1 & 0 \end\right) + c \left(\begin 0 & 0 & \;\; 1 & 0 \\ 0 & 0 & 0 & \;\; 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end\right) + d \left(\begin 0 & 0 & 0 & \;\; 1 \\ 0 & 0 & -1 & 0 \\ 0 & \;\; 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end\right)

In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the absolute value of a quaternion is the determinant of the corresponding matrix.

Cayley-Dickson construction

According to the Cayley-Dickson construction, a quaternion is an ordered pair of complex numbers. Letting j be a new root of −1, different from both i and −i, and given u and v are a pair of complex numbers, then

q = u + j v \,

is a quaternion.

If u = a + ib and v = c + id, then

q = a + i b + j c + j i d \,.

Moreover, let

j i = - i j \,,

so that

q = a + i b + j c + i j (-d) \,,

and also let the product of quaternions be associative.

With these rules, we can now derive the multiplication table for i, j and ij, the imaginary components of a quaternion:

i i = -1, \,

i j = (i j), \,

i (i j) = (i i) j = -j, \,

j i = - (i j), \,

j j = -1, \,

j (i j) = - j (j i) = - (j j) i = i, \,

(i j) i = - (j i) i = -j (i i) = j, \,

(i j) j = i (j j) = -i, \,

(i j) (i j) = -(i j) (j i) = -i (j j) i = i i = -1. \,

Notice how the dyad ij behaves just like the k in the definition.

For any complex number v = c + id, its product with j has the following property:

j v = v^* j \,

since

j v = j c + j i d = j c - (i j) d = (c - i d) j = v^* j \,.

Let p be the quaternion with complex components w and z:

p = w + j z \,.

Then the product qp is

q p = (u + j v) (w + j z) = u w + u j z + j v w + j v j z \,

= u w + j u^* z + j v w + j j v^* z \,

= (u w - v^* z) + j (u^* z + v w). \,

Since the product of complex numbers is commutative, we have

(u + j v) (w + j z) = (u w - z v^*) + j (u^* z + w v) \,

which is precisely how quaternion multiplication is defined by the Cayley-Dickson construction.

Note that if u = a + ib, v = c + id, and p = a + ib + jc + kd then p′s construction from u and v is rather

p = u + v j = u + j v^* \,.

H as a union of complex planes

Informal Introduction

There exists an intriguing way of understanding H that links its structure closely to the surface of an ordinary sphere of radius 1. In mathematics such a sphere is called a unit 2-sphere to emphasize that only its two-dimensional surface is being considered.

The first step is to translate the XYZ coordinates of the unit 2-sphere into the ijk coordinate system of quaternions, keeping the scalar (first) value of the quaternions set to zero. For example, the XYZ point ' becomes the quaternion 0 + 1i + 0j + 0k''. Since quaternion absolute lengths are calculated in the same way as XYZ radii, the resulting unit 2-sphere quaternions also all have absolute lengths (radii) of 1.

A less intuitive property of unit 2-sphere quaternions is that their squares all equal -1. This is true by definition for the three main axes of i, j, and k, but it can also be verified easily by trial for any arbitrary unit 2-sphere quaternion.

Since a length of 1 and a square of -1 are the defining properties of i, these unit 2-sphere quaternions look suspiciously like mathematical analogs to i. Furthermore, since each such quaternion has an "unused" scalar value associated with it, a fascinating conjecture becomes possible:

For any given ijk-only point on the quaternion unit 2-sphere, does the set of all quaternions that can be expressed as the sum of a real number and a multiple of that ijk-only point behave like a complex plane?

Somewhat unsurprisingly, the answer is yes.

That is, H can be partitioned in such a way that it looks like an infinite set of complex planes. Each such plane has its own unique version of i, although they all share the same real (scalar) axis. Furthermore, each unique i value corresponds to and is fully defined by a point on the surface of an ordinary unit-radius sphere, thus providing a strong connection between the geometry of ordinary spheres and the far less intuitive four-dimensional properties of H. Once a point on the unit 2-sphere has been selected, there is no mathematical difference in the behavior of the resulting subset of H and the more traditional concept of a single abstract complex plane.

Thus quaternions do not just extend the concept of i just to the two new axes j and k. They generalize i to an infinite set of points that happen to be the same ones found on the surface of an ordinary unit-radius sphere!

A more precise mathematical profile of how H can be interpreted as a union of complex planes is provided below.

Detailed Specification

Isomorphisms to the imaginary unit

The set of quaternions of absolute length (radius) 1 has the form of a 3-sphere or hypersphere, which is also called S³. Within this hypersphere there exists a subset of quaternions with the additional property that their squares are equal to −1. This subset has the geometric form of an ordinary sphere, or 2-sphere (S²). It can be understood as a three-dimensional "slice" of the larger hypersphere in much the same way that a circle is a two-dimensional "slice" of an ordinary sphere. For reasons explained below, this sphere-like subset of H is referred to here as Hi, where the i subscript refers to the imaginary unit, or \sqrt.

Identification of imaginary-unit isomorphisms

Membership in Hi can be specified using set notation. Two such tests are:

H_i = \left\ = \left\

Hi quaternions can also be identified by looking at whether it is true both that their first (scalar) component a is zero, and that their remaining bi, cj, and dk components have a length of 1 if interpreted as a three-dimensional vector:

H_i = \left\ \,\!

Isomorphisms to the complex plane

A notable feature of Hi is that every element i_r \in H_i can be used to define a subset of H (the full set of all quaternions) that behaves identically to the complex plane. That is, for every element i_r \in H_i there exists a subset Cr of the full set of quaternions H that is isomorphic to the complex plane.

C_r = \left\ \,\!

This is the reason for using the subscript i to label the set Hi.

Quaternions as isomorphic complex numbers

The union of the complex planes generated by all elements of Hi is the set of all quaternions H. This means any quaternion can be expressed as an isomorphic complex number whose imaginary unit is associated with a point on the ordinary unit sphere.

That is, given a quaternion q = a + bi + cj + dk, the corresponding isomorphic imaginary unit can be calculated by normalizing the ijk portion (only) of the quaternion:

b_r = \|r\| = \sqrt

i_r = \frac = \frac

The isomorphic complex number equivalent qr of the original quaternion q then becomes:

q_r = a + b_r i_r = a + \|r\| i_r

Euler's Formula

Additionally, since the general point on a circle as given by Euler's formula:

e^= \cos + i \sin \,\!

The general point on the 3-sphere of all unit-length quaternions is:

e^ = \cos + i_r \sin \,\!

Where   i_r \in H_i\ ,   and   \sin(\theta) =\frac  .

Commutative subrings

Finally, the relationship of quaternions to each other within ir subplanes of H can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions p and q commute (p q = q p) only if they lie in the same ir complex subplane of H, the profile of H as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. Ian R. Porteous's book Clifford Algebras and the Classical Groups (Cambridge, 1995) describes this derivation in proposition 8.13 on page 60.

Functions of a quaternion variable

Functions of a complex variable can be extended to functions of a quaternion variable as follows:

Let the complex function be written

f(z) = u(x,y) + i\ v(x,y)\,\!

where u and v are real-valued functions of two real variables. According to the above profile, any quaternion can be written

q = a + b\ r,\ \ \ r^ = -1 \ .

Then the extension is given by f(q) = u(a,b) + r\ v(a,b) \,\!.

This is called Fueter's method.

Three-dimensional and four-dimensional rotation groups

As is explained in more detail in quaternions and spatial rotation, the multiplicative group of non-zero quaternions acts by conjugation on the copy of R³ consisting of quaternions with real part equal to zero. The conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(t) is a rotation by an angle 2t, the axis of the rotation being the direction of the imaginary part. The advantages of quaternions are:

1. Non singular representation (compared with Euler angles for example)
2. More compact (and faster) than matrices
3. Pairs of unit quaternions represent a rotation in 4D space (see SO(4): Algebra of 4D rotations).

The set of all unit quaternions forms a 3-dimensional sphere S³ and a group (a Lie group) under multiplication. S³ is the double cover of the group SO(3,R) of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence.

The image of a subgroup of S³ is a point group, and conversely, the preimage of a point group is a subgroup of S³. 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 group S³ is isomorphic to SU(2), the group of complex unitary 2×2 matrices of determinant 1. Let A be the set of quaternions of the form a + bi + cj + dk where 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 and a lattice. There are 24 unit quaternions in this ring, and they are the vertices of a 24-cell regular polytope with Schläfli symbol .

Generalizations

If F is any field with characteristic different from 2, and a and b are elements of F, one may define a four-dimensional unitary associative algebra over F with basis 1, i, j, and ij, where i2 = a, j2 = b and ij = −ji (so ij2 = −ab). These algebras are called quaternion algebras and are isomorphic to the algebra of 2×2 matrices over F or form division algebras over F, depending on the choice of a and b.

History

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. He could not do so for 3 dimensions, but 4 dimensions produce quaternions. According to the story Hamilton told, on October 16, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

i^2 = j^2 = k^2 = ijk = -1\,

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). This involved abandoning the commutative law, a radical step for the time. Vector algebra and matrices had yet to be developed.

Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered quadruple (4-tuple) of one real number and three mutually orthogonal imaginary units with real coefficients, and described the first coordinate as the 'scalar' part, and the remaining three as the 'vector' part. If two quaternions with zero scalar parts are multiplied, the scalar part of the product is the negative of the dot product of the vector parts, while the vector part of the product is the cross product. But the significance of these was still to be discovered. Hamilton proceeded to popularize quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death.

Reading works written before 1900 on the subject of Classical Hamiltonian quaternions is difficult for modern readers because the notation used by early writers on the subject of quaternions, mostly based on the notation and vocabulary of Hamilton is different than what is used today.

The quaternions formed the theme for one of the first international mathematical associations, the Quaternion Society (1899 - 1913).

Recent years

Quaternions are often used in computer graphics (and associated geometric analysis) to represent rotations (see quaternions and spatial rotation) and orientations of objects in three-dimensional space. Certain fractals can plot in quaternion coordinates. They are smaller than other representations such as matrices, and operations on them such as composition can be computed more efficiently. Quaternions also see use in control theory, signal processing, attitude control, physics, bioinformatics (see: Root mean square deviation (bioinformatics)), and orbital mechanics, mainly for representing rotations/orientations in three dimensions. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations. There is also less overhead in using quaternions compared to using rotation matrices, because a quaternion has only four components instead of nine, so the multiplication algorithms to combine successive rotations are faster, and the result is much easier to renormalize afterwards.

Since 1989, the Department of Mathematics of the National University of Ireland, Maynooth has organized a pilgrimage, where scientists (including physicists Murray Gell-Mann in 2002 and Steven Weinberg in 2005 and mathematician Andrew Wiles in 2003) take a walk from Dunsink Observatory to the Royal Canal bridge where, unfortunately, no trace of Hamilton's carving remains.

Quotes

• "I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc." — William Rowan Hamilton (ed. Quoted in a letter from Tait to Cayley.)
• "Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in R.P. Graves, "Life of Sir William Rowan Hamilton")
• "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." — Lord Kelvin, 1892.
• "Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in every-day life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols." — Ludwik Silberstein, preparing the second edition of his Theory of Relativity in 1924
• "… quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist." — Simon L. Altmann, 1986

Quaternions in fiction

• "...the thing about a Quaternion 'is' is that we're obliged to encounter it in more than one guise. As a vector quotient. As a way of plotting complex numbers along three axes instead of two. As a list of instructions for turning one vector into another..... And considered subjectively, as an act of becoming longer or shorter, while at the same time turning, among axes whose unit vector is not the familiar and comforting 'one' but the altogether disquieting square root of minus one. If you were a vector, mademoiselle, you would begin in the 'real' world, change your length, enter an 'imaginary' reference system, rotate up to three different ways, and return to 'reality' a new person. Or vector....." — Thomas Pynchon, in Against the Day, 2006, p. 534, a fictional conversation, overheard by his fictional characters Kit Traverse and Umeki Tsurigane, at a fictional gathering of "Quaternionnaires from around the globe", in Ostend, Belgium in or around the year 1905.

Recent developments and research directions

Quaternions and Minkowski metric

As a linear algebra over the reals, quaternions constitute a real vector space with a rank three tensor S on it, sometimes called the structure tensor. This once contravariant twice covariant tensor converts a one-form t and vectors a and b to a real number S(t, a, b). For each one-form t, S is a twice covariant tensor, which, if symmetric, is an inner product on H. Since any real vector space can also be considered a linear manifold, such an inner product is naturally extended to a tensor field, and in case of its nondegeneracy, becomes a (pseudo- or proper-)Euclidean metric . For quaternions this inner product is indefinite, its signature is independent of the one-form t, and the corresponding pseudo-Euclidean metric is Minkowski [1]. This metric is automatically extended over the Lie group of nonzero quaternions along its left invariant vector fields resulting in a closed FLRW metric [2] – an important solution of the Einstein equations. These results have some implications for the problem of compatibility between quantum mechanics and general relativity within the framework of quantum gravity [3].

See also

• Quaternion group
• Split-quaternion (Coquaternion)
• 3-sphere
• SO(4)
• Associative algebra
• Complex number
• Division algebra
• Dual quaternion
• Geometric algebra
• Hypercomplex number
• Musean hypernumber
• Octonion
• Quaternions and spatial rotation
• Biquaternion
• Hyperbolic quaternion
• Tesseract
• Hurwitz quaternion
• Euler Angles
• Clifford algebra
• Exterior algebra
• Slerp

External articles and resources

Books and publications

• Hamilton, William Rowan (1853), "Lectures on Quaternions". Royal Irish Academy.
• Tait, Peter Guthrie (1873), "An elementary treatise on quaternions". 2d ed., Cambridge, [Eng.] : The University Press.
• Maxwell, James Clerk (1873), "A Treatise on Electricity and Magnetism". Clarendon Press, Oxford.
• Tait, Peter Guthrie (1886), "Quaternion". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 160-164. (bzipped PostScript file)
• Joly, Charles Jasper (1905), "A manual of quaternions". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84
• Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
• 1911 encyclopedia: "Quaternion".
• Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3, pp207-220, MathSciNet.
• Du Val, Patrick (1964), "Homographies, quaternions, and rotations". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81
• Crowe, Michael J. (1967), "A History of Vector Analysis: The Evolution of the Idea of a Vectorial System". University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, MacFarlane, MacAuley, Gibbs, Heaviside). The competition between quaternions and other systems is a major theme.
• Altmann, Simon L. (1986), "Rotations, quaternions, and double groups". Oxford [Oxfordshire] : Clarendon Press ; New York : Oxford University Press. LCCN 85013615 ISBN 0-19-855372-2
• Adler, Stephen L. (1995), "Quaternionic quantum mechanics and quantum fields". New York : Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X
• Trifonov, Vladimir (1995), "A Linear Solution of the Four-Dimensionality Problem", Europhysics Letters, 32 (8) 621-626, DOI: 10.1209/0295-5075/32/8/001
• Ward, J. P. (1997), "Quaternions and Cayley Numbers: Algebra and Applications", Kluwer Academic Publishers. ISBN 0-7923-4513-4
• Kantor, I. L. and Solodnikov, A. S. (1989), "Hypercomplex numbers, an elementary introduction to algebras", Springer-Verlag, New York, ISBN 0-387-96980-2
• Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "Quaternionic and Clifford calculus for physicists and engineers". Chichester ; New York : Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 ISBN 0-471-96200-7
• Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8
• Conway, John Horton, and Smith, Derek A. (2003), "On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry", A. K. Peters, Ltd. ISBN 1-56881-134-9 (review).
• Kravchenko, Vladislav (2003), "Applied Quaternionic Analysis", Heldermann Verlag ISBN 3-88538-228-8.
• Hanson, Andrew J. (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
• Trifonov, Vladimir (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251-257, DOI: 10.1007/s10773-006-9234-9

Links and monographs

• Matrix and Quaternion FAQ v1.21 Frequently Asked Questions
• Geometric Tools documentation Includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
• Patrick-Gilles Maillot Provides free Fortran and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
• Geometric Tools source code Includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
• Doing Physics with Quaternions
• Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
• The Physical Heritage of Sir W. R. Hamilton (PDF)
• Hamilton’s Research on Quaternions
• Quaternion Julia Fractals 3D Raytraced Quaternion Julia Fractals by David J. Grossman
• Quaternion Math and Conversions Great page explaining basic math with links to straight forward rotation conversion formulae.
• John H. Mathews, Bibliography for Quaternions.
• Quaternion powers on GameDev.net
• Andrew Hanson, Visualizing Quaternions home page.
• Representing Attitude with Euler Angles and Quaternions: A Reference, Technical report and Matlab toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.
• Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations., arXiv General Mathematics 2005.
• Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007.
• NUI Maynooth Department of Mathematics, Hamilton Walk.
• OpenGL:Tutorials:Using Quaternions to represent rotation
• D. Erickson, Derivation of rotation matrix from unitary quaternion representation in old paper:http://aiss.suffield.drdc-rddc.gc.ca/uploads/quaternion.pdf
• Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent", https://webspace.utexas.edu/aam829/1/m/NegativeMath.html

Software

• Euler Quaternion Pro A free GUI based utility that converts Euler angles to Quaternions around X,Y and Z (roll, pitch and yaw) axis and performs conjugate, addition, subtraction, multiplication, great circle interpolation operations on converted Quaternions.
• Quaternion Calculator [Java]
• Quaternion Toolbox for Matlab
• Boost library support for Quaternions in C++
• Mathematics of flight simulation >Turbo-PASCAL software for quaternions, Euler angles and Extended Euler angles