Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = -k, ij = -ji

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 Although they have been superseded in most applications by vectors and matrices, they still find uses in both theoretical and applied mathematics, in particular for calculations involving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. The term Quaternion can have several possible meanings: A mathematical entity A musical composition by Sofia Gubaidulina A secret honorary society at Furman University in Greenville, South Carolina This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Year 1843 (MDCCCXLIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Friday of the 12-day slower Julian calendar). ... For other uses, see Mechanic (disambiguation). ... In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... This article is about vectors that have a particular relation to the spatial coordinates. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ... This article is about the scientific discipline of 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 (Unicode ℍ). It can also be given the Clifford algebra classifications Cℓ_0,2(R) = Cℓ⁰_3,0(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. In mathematics, a normed division algebra A is a division algebra over the real or complex numbers which is also a normed vector space, with norm || . || satisfying ||xy|| = ||x|| ||y|| for all x and y in A. While the definition allows normed division algebras to be infinite-dimensional, this, in... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... An example of blackboard bold letters. ... In mathematics, Clifford algebras are a type of associative algebra. ... In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite dimensional associative division algebras over the real numbers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... Please refer to Real vs. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...

Definition

The quaternions are defined as the ring: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

where addition is defined by:

and multiplication is defined by expanding:

using the distributive law and then applying the defining relations:

Every quaternion is a unique and real linear combination of the basis quaternions 1, i, j, and k. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

Properties

Basis multiplication

The set of equations

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

For example, since

right-multiplying both sides by k gives

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 z² + 1 = 0, for instance, has infinitely many quaternion solutions z = bi + cj + dk with b² + c² + d² = 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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

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. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another 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, Q₈. Times table redirects here. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...

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. In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, associativity is a property that a binary operation can have. ...

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. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...

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. Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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.

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 , such that the 1 would be in the t location.

The vector form of a quaternion may also be used. This form assumes that .

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

let

let

Antiautomorphisms

Negation (Additive inverse)

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

Conjugation (Spatial inverse)

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. Clifford algebras are a type of associative algebra in mathematics. ...

NOTE: The operator symbol for the conjugate is not standardized. This can sometimes be seen as , , , , and sometimes other symbols are used. Later in this article, is used to denote the conjugate.

Common binary operations

Addition

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

Subtraction

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

Quaternion products

Grassmann product

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... A mathematical operator (typically a binary operator, represented by *) is anticommutative iff it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ...

let

The components of Q:

It should be noted at this point that the anticommutative part of the product is the cross product of the vectors . 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. For the cross product in algebraic topology, see Künneth theorem. ...

Properties:

• Non-commutative: for some and , .
• Associative:
• Left and Right Distributive:

Inner product

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.

Example:

In terms of the Grassmann product:

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

Properties:

• Commutative:
• Associative:
• Distributive:

Outer-product

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

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

and the absolute value of p is the non-negative real number defined by In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

where p ^* is the conjugate of p, or a² − b² − c² − d².

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. For distance between people, see proxemics. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

Given quarternions

with

some other products are defined as follows.

Quaternion cross-product

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:

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

Quaternion even-product

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.

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

Quaternion Euclidean product

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

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

When p = q, the result is the conjugate.

Quaternion reciprocal

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:

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.

Quaternion division

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.

Quaternion scalar

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

Quaternion vector

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:

Quaternion modulus

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

Quaternion sign

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: In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...

Quaternion argument

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

Example

Let

Then

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. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... This article gives an overview of the various ways to perform matrix multiplication. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

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

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 In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. ...

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. In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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 In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ...

is a quaternion.

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

.

Moreover, let

,

so that

,

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:

Notice how the dyad ij behaves just like the k in the definition. A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i. ...

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

since

.

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

.

Then the product qp is

Since the product of complex numbers is commutative, we have

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

.

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. For other uses, see sphere (disambiguation). ...

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 <1,0,0> 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 surprisingly, the answer is yes. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

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 H_i, where the i subscript refers to the imaginary unit, or . In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ... For other uses, see sphere (disambiguation). ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

Identification of imaginary-unit isomorphisms

Membership in H_i can be specified using set notation. Two such tests are: Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

H_i 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:

Isomorphisms to the complex plane

A notable feature of H_i is that every element 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 there exists a subset C_r of the full set of quaternions H that is isomorphic to the complex plane. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

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

Quaternions as isomorphic complex numbers

The union of the complex planes generated by all elements of H_i 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:

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

Euler's Formula

Additionally, since the general point on a circle as given by Euler's formula: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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

Where     and     .

Commutative subrings

Finally, the relationship of quaternions to each other within i_r 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 i_r 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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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

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

.

Then the extension is given by .

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: Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ...

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. Euler angles are a means of representing the spatial orientation of an object. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... For other uses, see Fourth dimension (disambiguation). ... SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

For more details on this topic, see Point groups in three dimensions#Spin_analogs.

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. A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... A discrete point group in 3D is a finite symmetry group in 3D that leaves the origin fixed. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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 {3,4,3}. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. ...

Generalizations

Main article: quaternion algebra

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. In mathematics, a quaternion algebra over a field L is a particular kind of central simple algebra A over L, namely such an algebra that has dimension 4, and therefore becomes the 2×2 matrix algebra over some field extension of L, by extending scalars. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ...

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 For other persons named William Hamilton, see William Hamilton (disambiguation). ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... This article is about the mathematical construct. ... For other uses, see Dublin (disambiguation). ...

Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for quaternion multiplication
i² = j² = k² = i j k = −1
& cut it on a stone of this bridge.

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. A plaque on Broom Bridge in Dublin, commemorating Hamiltons invention of Quaternions My father took the picture and agrees to transfer the copyright to me. ... A plaque on Broom Bridge in Dublin, commemorating Hamiltons invention of Quaternions My father took the picture and agrees to transfer the copyright to me. ... For other uses, see Dublin (disambiguation). ... Broom Bridge Plaque on Broom Bridge Broom Bridge, also known as Brougham Bridge, is a small bridge along Broombridge road which crosses the Royal Canal in Dublin, Ireland. ...

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. For the cross product in algebraic topology, see Künneth theorem. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... For the cross product in algebraic topology, see Künneth theorem. ...

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. This article is about the scientific discipline of computer graphics. ... Quaternions provide a convenient mathematical notation for representing orientations and rotations of objects. ... 2-dimensional renderings (ie. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... For control theory in psychology and sociology, see control theory (sociology). ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... // In the context of spacecraft, attitude control is control of the angular position and rotation of the spacecraft, either relative to the object that it is orbiting, or relative to the celestial sphere. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Map of the human X chromosome (from the NCBI website). ... The root mean square deviation (RMSD) is the measure of the average distance between the backbones of superimposed proteins. ... This article or section should be merged with Celestial Mechanics Astrodynamics is the study and creation of orbits, especially those of artificial satellites. ...

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. The National University of Ireland, Maynooth (NUIM) was founded in 1997 by the Universities Act, 1997 as a constituent university of the National University of Ireland. ... Murray Gell-Mann (born September 15, 1929 in Manhattan, New York City, USA) is an American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles. ... Steven Weinberg (born May 3, 1933) is an American physicist. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... The Dunsink Observatory is an astronomical observatory established in approximately 1785 near the city of Dublin, Ireland. ...

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

James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... William Thomson, Archbishop of York, has the same name as this man. ... Ludwik Silberstein (1872 – 1948) was a Polish-American physicist that helped make special relativity and general relativity staples of university coursework. ... This article or section may contain original research or unverified claims. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense