Quaternion


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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
Quaternions are generally represented in the form:
where, and are real numbers, and , and are the fundamental quaternion units.
Quaternions are used in pure mathematics, and also have practical uses in applied mathematics—in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by , or in blackboard bold by . It can also be given by the Clifford algebra classifications. The algebra holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the reals.
The unit quaternions can be thought of as a choice of a group structure on the 3-sphere that gives the group Spin, which is isomorphic to SU and also to the universal cover of SO.

History

Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity and Olinde Rodrigues' parameterization of general rotations by four parameters, but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.
Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space.
The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions,
into the stone of Brougham Bridge as he paused on it. Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery.
On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery. This letter was later published in a letter to the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science; Hamilton states:
Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long; it was edited by his son and published shortly after his death.
After Hamilton's death, his student Peter Tait continued promoting quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described entirely in terms of quaternions. There was even a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems.
From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.
However, quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles, they are not susceptible to “gimbal lock”. For this reason, quaternions are used in computer graphics, computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.

Quaternions in physics

P.R. Girard's 1984 essay The quaternion group and modern physics discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely, the Lorentz group, the general theory of relativity group, the Clifford algebra and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion. Next he used complex quaternions to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions as an instance of Clifford algebra. Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions.
The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720°., their use has not overtaken rotation groups.

Definition

A quaternion is an expression of the form
where,,,, are real numbers, and,,, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of,,, is 0, the corresponding term is omitted; if,,, are all zero, the quaternion is the zero quaternion, denoted 0; if one of,, equals 1, the corresponding term is written simply, or.
Hamilton describes a quaternion, as consisting of a scalar part and a vector part. The quaternion is called the vector part of, and is the scalar part of. A quaternion that equals its real part is called a scalar or real quaternion, and is identified with the corresponding real number. That is, the real numbers are embedded in the quaternions. A quaternion that equals its vector part is called a vector quaternion.
The set of quaternions is made a 4 dimensional vector space over the real numbers, with as a basis, by the componentwise addition
and the componentwise scalar multiplication
A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:
Thus the quaternions form a division algebra.

Multiplication of basis elements

The basis elements, and commute with the real quaternion 1, that is
The other products of basis elements are defined by
and
These multiplication formulas are equivalent to
In fact, the equality results from
The converse implication results from manipulations similar to the following. By right-multiplying both sides of by, one gets
All other products can be determined by similar methods.

Center

The center of a noncommutative ring is the subring of elements such that for every. The center of the quaternion algebra is the subfield of real quaternions. In fact, it is a part of the definition that the real quaternions belong to the center. Conversely, if belongs to the center, then
and. A similar computation with instead of shows that one has also. Thus is a real quaternion.
The quaternion form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that a polynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equation has infinitely many quaternion solutions, which are the quaternions such that. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.

Hamilton product

For two elements and, their product, called the Hamilton product , is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:
Now the basis elements can be multiplied using the rules given above to get:
The product of two rotation quaternions will be equivalent to the rotation followed by the rotation

Scalar and vector parts

A quaternion of the form, where is a real number, is called scalar, and a quaternion of the form, where, and are real numbers, and at least one of or is nonzero, is called a vector quaternion. If is any quaternion, then is called its scalar part and is called its vector part. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space.
Hamilton also called vector quaternions right quaternions and real numbers scalar quaternions.
If a quaternion is divided up into a scalar part and a vector part, i.e.
then the formulas for addition and multiplication are:
where "·" is the dot product and "×" is the cross product.

Conjugation, the norm, and reciprocal

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition of elements of Clifford algebras. To define it, let be a quaternion. The conjugate of is the quaternion. It is denoted by, qt,, or. Conjugation is an involution, meaning that it is its own inverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if and are quaternions, then, not.
The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of is and the vector part of is
The square root of the product of a quaternion with its conjugate is called its norm and is denoted . In formula, this is expressed as follows:
This is always a non-negative real number, and it is the same as the Euclidean norm on considered as the vector space. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if is real, then
This is a special case of the fact that the norm is multiplicative, meaning that
for any two quaternions and. Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity
and hence from the multiplicative property of determinants of square matrices.
This norm makes it possible to define the distance between and as the norm of their difference:
This makes a metric space. Addition and multiplication are continuous in the metric topology. Indeed, for any scalar, positive it holds
Continuity follows from taking to zero in the limit. Continuity for multiplication holds similarly.

Unit quaternion

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion by its norm produces a unit quaternion called the versor of :
Every quaternion has a polar decomposition.
Using conjugation and the norm makes it possible to define the reciprocal of a non-zero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of and is 1. So the reciprocal of is defined to be
This makes it possible to divide two quaternions and in two different ways. That is, their quotient can be either or ; in general, those products are different, depending on the order of multiplication, except for the special case that and are scalar multiples of each other. Hence, the notation is ambiguous because it does not specify whether divides on the left or the right.

Algebraic properties

The set of all quaternions is a vector space over the real numbers with dimension 4. have dimension 1, the complex numbers have dimension 2, and the octonions have dimension 8.) Multiplication of quaternions is associative and distributes over vector addition, but it is not commutative. Therefore, the quaternions are a non-commutative associative algebra over the real numbers. Even though contains copies of the complex numbers, it is not an associative algebra over the complex numbers.
Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication. Finite-dimensional associative division algebras over the real numbers are very rare. The Frobenius theorem states that there are exactly three:,, and. The norm makes the quaternions into a normed algebra, and normed division algebras over the reals are also very rare: Hurwitz's theorem says that there are only four:,,, and . The quaternions are also an example of a composition algebra and of a unital Banach algebra.
Because the product of any two basis vectors is plus or minus another basis vector, the set forms a group under multiplication. This non-abelian group is called the quaternion group and is denoted. The real group ring of is a ring which is also an eight-dimensional vector space over. It has one basis vector for each element of. The quaternions are isomorphic to the quotient ring of by the ideal generated by the elements,, and. Here the first term in each of the differences 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 the geometry of

The vector part of a quaternion can be interpreted as a coordinate vector in ; therefore, the algebraic operations of the quaternions reflect the geometry of. 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.
For the remainder of this section,,, and will denote both the three imaginary basis vectors of and a basis for. 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, is
It can also be expressed in a component-free manner as
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
This is equal to the vector part of the product , as well as the vector part of. It also has the formula
For the commutator,, of two vector quaternions one obtains
In general, let and be quaternions and write
where and are the scalar parts, and and are the vector parts of and. Then we have the formula
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 and were called versors by Hamilton. Also see Quaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.
See Hanson 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. In the terminology of abstract algebra, these are injective homomorphisms from to the matrix rings and respectively.
Using 2 × 2 complex matrices, the quaternion can be represented as
This representation has the following properties:
Using 4 × 4 real matrices, that same quaternion can be written as
However, the representation of quaternions in is not unique. For example, the same quaternion can also be represented as
There exist 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. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by setting
Each 4×4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table
×ad−b−c
aad−b−c
−d−dac−b
bb−ca−d
ccbda

which is isomorphic — through — to
×1k−i−j
11k−i−j
−k−k1j−i
ii−j1−k
jjik1

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, 2 possible choices for the third column, and 1 possible choice for the fourth column ; 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 be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements and. A vector in can be written in terms of the basis elements and as
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 as ordered pairs and quaternions as quadruples, then the correspondence is

Square roots of −1

In the complex numbers,, there are just two numbers, i and −i, whose square is −1 . In there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in. To see this, let be a quaternion, and assume that its square is −1. In terms of, and, this means
To satisfy the last three equations, either or, and are all 0. The latter is impossible because a is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. 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.

as a union of complex planes

Each pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. If then the copy is determined by the function
In the language of abstract algebra, each is an injective ring homomorphism from to. The images of the embeddings corresponding to and − are identical.
Every non-real quaternion determines a planar subspace in that is isomorphic to : Write as the sum of its scalar part and its vector part:
Decompose the vector part further as the product of its norm and its versor:
The versor of the vector part of,, is a right versor with –1 as its square. Therefore, it determines a copy of the complex numbers by the function
Under this function, is the image of the complex number. Thus is the union of complex planes intersecting in a common real line, where the union is taken over the sphere of square roots of minus one, bearing in mind that the same plane is associated with any pair of antipodal points on the sphere of right versors.

Commutative subrings

The relationship of quaternions to each other within the complex subplanes of can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions and commute only if they lie in the same complex subplane of, the profile of as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. This method of commutative subrings is also used to profile the split-quaternions, which as an algebra over the reals are isomorphic to 2 × 2 real matrices.

Functions of a quaternion variable

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

Given a quaternion,
the exponential is computed as
and the logarithm is
It follows that the polar decomposition of a quaternion may be written
where the angle
and the unit vector is defined by:
Any unit quaternion may be expressed in polar form as.
The power of a quaternion raised to an arbitrary exponent is given by:

Geodesic norm

The geodesic distance between unit quaternions and is defined as:
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:

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 non-zero quaternion. All elements that are conjugate to a given element have the same real part and the same norm of the vector part.
Thus the multiplicative group of non-zero quaternions acts by conjugation on the copy of consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion 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:
The set of all unit quaternions forms a 3-sphere and a group under multiplication, double covering the group of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions correspond to every rotation under the above correspondence. See the 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 2×2 matrices of determinant 1.
Let be the set of quaternions of the form where and are either all integers or all rational numbers with odd numerator and denominator 2. The set is a ring 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 .
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 of the Clifford algebra. This is an associative multivector algebra built up from fundamental basis elements,, using the product rules
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:
Two reflections make a rotation by an angle twice the angle between the two reflection planes, so
corresponds to a rotation of 180° in the plane containing σ1 and σ2. This is very similar to the corresponding quaternion formula,
In fact, the two are identical, if we make the identification
and it is straightforward to confirm that this preserves the Hamilton relations
In this picture, so-called "vector quaternions" correspond not to vectors but to bivectors – quantities with magnitude 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, 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:
For further detail about the geometrical uses of Clifford algebras, see Geometric algebra.

Brauer group

The quaternions are "essentially" the only central simple algebra over the real numbers, in the sense that every CSA over the reals is Brauer equivalent to either the reals or the quaternions. Explicitly, the Brauer group of the reals consists of two classes, represented by the reals 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, CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals.
CSAs – rings over a field, which are simple algebras 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 reals may be compared with the fact that the complex numbers are the only non-trivial field extension of the reals.

Quotations

Books and publications