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,
(1)
|
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
(2)
|
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
(3)
|
where
and
are complex
numbers,
,
,
, and
are real, and
is the complex conjugate
of
.
Quaternions can also be represented using the complex matrices
(4)
| |||
(5)
| |||
(6)
| |||
(7)
|
(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
(8)
| |||
(9)
| |||
(10)
|
Therefore ,
, and
are three essentially different solutions of the matrix equation
(11)
|
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
(12)
| |||
(13)
| |||
(14)
| |||
(15)
|
The quaternions satisfy the following identities, sometimes known as Hamilton's rules,
(16)
| |
(17)
| |
(18)
| |
(19)
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They have the following multiplication table.
1 | ||||
1 | 1 | |||
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
(20)
|
The quaternion conjugate is given by
(21)
|
The sum of two quaternions is then
(22)
|
and the product of two quaternions is
(23)
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The quaternion norm is therefore defined by
(24)
|
In this notation, the quaternions are closely related to four-vectors.
Quaternions can be interpreted as a scalar plus a vector by writing
(25)
|
where .
In this notation, quaternion multiplication has the particularly simple form
(26)
| |||
(27)
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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
(28)
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and the norm is multiplicative
(29)
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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
(30)
|
(Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called Euler parameters. After rotation, a point is then given by
(31)
|
since .
A concatenation of two rotations, first
and then
, can be computed using the identity
(32)
|
(Goldstein 1980).