The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he was walking along the Royal Canal on his way to a meeting of the Irish Academy, and Hamilton was so pleased with his discovery that he scratched the fundamental formula of quaternion algebra,
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into the stone of the Brougham bridge (Mishchenko and Solovyov 2000). The set of quaternions is denoted , , or , and the quaternions are a single example of a more general class of hypercomplex numbers discovered by Hamilton. While the quaternions are not commutative, they are associative, and they form a group known as the quaternion group.
By analogy with the complex numbers being representable as a sum of real and imaginary parts, , a quaternion can also be written as a linear combination
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The quaternion is implemented as Quaternion[a, b, c, d] in the Wolfram Language package Quaternions` where however , , , and must be explicit real numbers. Note also that NonCommutativeMultiply (i.e., **) must be used for multiplication of these objects rather than usual multiplication (i.e., *).
A variety of fractals can be explored in the space of quaternions. For example, fixing gives the complex plane, allowing the Mandelbrot set. By fixing or at different values, three-dimensional quaternionic fractals have been produced (Sandin et al. , Meyer 2002, Holdaway 2006).
The quaternions can be represented using complex matrices
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where and are complex numbers, , , , and are real, and is the complex conjugate of .
Quaternions can also be represented using the complex matrices
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(Arfken 1985, p. 185). Note that here is used to denote the identity matrix, not . The matrices are closely related to the Pauli matrices , , and combined with the identity matrix.
From the above definitions, it follows that
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Therefore , , and are three essentially different solutions of the matrix equation
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which could be considered the square roots of the negative identity matrix. A linear combination of basis quaternions with integer coefficients is sometimes called a Hamiltonian integer.
In , the basis of the quaternions can be given by
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The quaternions satisfy the following identities, sometimes known as Hamilton's rules,
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They have the following multiplication table.
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The quaternions , , , and form a non-Abelian group of order eight (with multiplication as the group operation).
The quaternions can be written in the form
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The quaternion conjugate is given by
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The sum of two quaternions is then
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and the product of two quaternions is
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The quaternion norm is therefore defined by
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In this notation, the quaternions are closely related to four-vectors.
Quaternions can be interpreted as a scalar plus a vector by writing
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where . In this notation, quaternion multiplication has the particularly simple form
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Division is uniquely defined (except by zero), so quaternions form a division algebra. The inverse (reciprocal) of a quaternion is given by
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and the norm is multiplicative
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In fact, the product of two quaternion norms immediately gives the Euler four-square identity.
A rotation about the unit vector by an angle can be computed using the quaternion
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(Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called Euler parameters. After rotation, a point is then given by
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since . A concatenation of two rotations, first and then , can be computed using the identity
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(Goldstein 1980).