Noncommutative extension of the complex numbers
Quaternion multiplication table
↓ × →
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1
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i
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j
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k
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1
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1
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i
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j
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k
|
i
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i
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?1
|
k
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?
j
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j
|
j
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?
k
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?1
|
i
|
k
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k
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j
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?
i
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?1
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Left column shows the left factor, top row shows the right factor. Also,
and
for
,
.
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In
mathematics
, the
quaternion
number system
extends the
complex numbers
. Quaternions were first described by the Irish mathematician
William Rowan Hamilton
in 1843
[1]
[2]
and applied to
mechanics
in
three-dimensional space
. The algebra of quaternions is often denoted by
H
(for
Hamilton
), or in
blackboard bold
by
Quaternions are not a field, because multiplication of quaternions is not, in general,
commutative
. Quaternions provide a definition of the quotient of two
vectors
in a three-dimensional space.
[3]
[4]
Quaternions are generally represented in the form
where the coefficients
a
,
b
,
c
,
d
are
real numbers
, and
1,
i
,
j
,
k
are the
basis vectors
or
basis elements
.
[5]
Quaternions are used in
pure mathematics
, but also have practical uses in
applied mathematics
, particularly for
calculations involving three-dimensional rotations
, such as in
three-dimensional computer graphics
,
computer vision
,
magnetic resonance imaging
[6]
and
crystallographic texture
analysis.
[7]
They can be used alongside other methods of rotation, such as
Euler angles
and
rotation matrices
, or as an alternative to them, depending on the application.
In modern terms, quaternions form a four-dimensional
associative
normed
division algebra
over the real numbers, and therefore a ring, also a
division ring
and a
domain
. It is a special case of a
Clifford algebra
,
classified
as
It was the first noncommutative division algebra to be discovered.
According to the
Frobenius theorem
, the algebra
is one of only two finite-dimensional
division rings
containing a proper
subring
isomorphic
to the real numbers; the other being the complex numbers. These rings are also
Euclidean Hurwitz algebras
, of which the quaternions are the largest
associative algebra
(and hence the largest ring). Further extending the quaternions yields the
non-associative
octonions
, which is the last
normed division algebra
over the real numbers. The next extension gives the
sedenions
, which have
zero divisors
and so cannot be a normed division algebra.
[8]
The
unit quaternions
give a
group
structure on the
3-sphere
S
3
isomorphic to the groups
Spin(3)
and
SU(2)
, i.e. the
universal cover
group of
SO(3)
. The positive and negative basis vectors form the eight-element
quaternion group
.
History
[
edit
]
Quaternions were introduced by Hamilton in 1843.
[9]
Important precursors to this work included
Euler's four-square identity
(1748) and
Olinde Rodrigues
'
parameterization of general rotations by four parameters
(1840), but neither of these writers treated the four-parameter rotations as an algebra.
[10]
[11]
Carl Friedrich Gauss
had also discovered quaternions in 1819, but this work was not published until 1900.
[12]
[13]
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, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact,
Ferdinand Georg Frobenius
later
proved
in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras:
(complex numbers) and
(quaternions) which have dimension 1, 2, and 4 respectively.
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
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
;
[14]
Hamilton states:
And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.
[14]
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
,
[15]
was 800 pages long; it was edited by
his son
and published shortly after his death.
After Hamilton's death, the Scottish mathematical physicist
Peter Tait
became the chief exponent of 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
,
[16]
[17]
computer vision
,
robotics
,
[18]
nuclear magnetic resonance
image sampling,
[6]
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
.
[19]
Quaternions in physics
[
edit
]
The finding of 1924 that in
quantum mechanics
the
spin
of an electron and other matter particles (known as
spinors
) can be described using quaternions (in the form of the famous Pauli spin matrices) furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "
Plate trick
").
[20]
[21]
As of 2018
[update]
, their use has not overtaken
rotation groups
.
[a]
Definition
[
edit
]
A
quaternion
is an
expression
of the form
where
a
,
b
,
c
,
d
, are real numbers, and
i
,
j
,
k
, are
symbols
that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one of
a
,
b
,
c
,
d
is 0, the corresponding term is omitted; if
a
,
b
,
c
,
d
are all zero, the quaternion is the
zero quaternion
, denoted 0; if one of
b
,
c
,
d
equals 1, the corresponding term is written simply
i
,
j
, or
k
.
Hamilton describes a quaternion
, as consisting of a
scalar
part and a vector part. The quaternion
is called the
vector part
(sometimes
imaginary part
) of
q
, and
a
is the
scalar part
(sometimes
real part
) of
q
. A quaternion that equals its real part (that is, its vector part is zero) 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. (More properly, the
field
of real numbers is isomorphic to a subset of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.)
[22]
A quaternion that equals its vector part is called a
vector quaternion
.
The set of quaternions is a 4-dimensional
vector space
over the real numbers, with
as a
basis
, by the component-wise addition
and the component-wise scalar multiplication
A multiplicative group structure, called the
Hamilton product
, denoted by juxtaposition, can be defined on the quaternions in the following way:
- The real quaternion
1
is the
identity element
.
- The
real
quaternions commute with all other quaternions, that is
aq
=
qa
for every quaternion
q
and every real quaternion
a
. In algebraic terminology this is to say that the field of real quaternions are the
center
of this quaternion algebra.
- The product is first given for the basis elements (see next subsection), and then extended to all quaternions by using the
distributive property
and the center property of the real quaternions. The Hamilton product is not
commutative
, but is
associative
, thus the quaternions form an associative algebra over the real numbers.
- Additionally, every nonzero quaternion has an inverse with respect to the Hamilton product:
Thus the quaternions form a division algebra.
Multiplication of basis elements
[
edit
]
Multiplication table
×
|
1
|
i
|
j
|
k
|
1
|
1
|
i
|
j
|
k
|
i
|
i
|
?1
|
k
|
?
j
|
j
|
j
|
?
k
|
?1
|
i
|
k
|
k
|
j
|
?
i
|
?1
|
Non commutativity is emphasized by colored squares
The multiplication with
1
of the basis elements
i
,
j
, and
k
is defined by the fact that
1
is a
multiplicative identity
, that is,
The products of other basis elements are
Combining these rules,
Center
[
edit
]
The
center
of a
noncommutative ring
is the subring of elements
c
such that
cx
=
xc
for every
x
. 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
q
=
a
+
b
i
+
c
j
+
d
k
belongs to the center, then
and
c
=
d
= 0
. A similar computation with
j
instead of
i
shows that one has also
b
= 0
. Thus
q
=
a
is a
real
quaternion.
The quaternions 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
z
2
+ 1 = 0
,
has infinitely many quaternion solutions, which are the quaternions
z
=
b
i
+
c
j
+
d
k
such that
b
2
+
c
2
+
d
2
= 1
. Thus these "roots of ?1" form a
unit sphere
in the three-dimensional space of vector quaternions.
Hamilton product
[
edit
]
For two elements
a
1
+
b
1
i
+
c
1
j
+
d
1
k
and
a
2
+
b
2
i
+
c
2
j
+
d
2
k
, their product, called the
Hamilton product
(
a
1
+
b
1
i
+
c
1
j
+
d
1
k
) (
a
2
+
b
2
i
+
c
2
j
+
d
2
k
), 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:
[9]
Scalar and vector parts
[
edit
]
A quaternion of the form
a
+ 0
i
+ 0
j
+ 0
k
, where
a
is a real number, is called
scalar
, and a quaternion of the form
0 +
b
i
+
c
j
+
d
k
, where
b
,
c
, and
d
are real numbers, and at least one of
b
,
c
, or
d
is nonzero, is called a
vector quaternion
. If
a
+
b
i
+
c
j
+
d
k
is any quaternion, then
a
is called its
scalar part
and
b
i
+
c
j
+
d
k
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
[b]
Hamilton also called vector quaternions
right quaternions
[24]
[25]
and real numbers (considered as quaternions with zero vector part)
scalar quaternions
.
If a quaternion is divided up into a scalar part and a vector part, that is,
then the formulas for addition, multiplication, and multiplicative inverse are
where "
" and "
" denote respectively the
dot product
and the
cross product
.
Conjugation, the norm, and reciprocal
[
edit
]
Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, let
be a quaternion. The
conjugate
of
q
is the quaternion
. It is denoted by
q
?
,
q
t
,
, or
q
.
[9]
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
p
and
q
are quaternions, then
(
pq
)
?
=
q
?
p
?
, not
p
?
q
?
.
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
p
is
1
/
2
(
p
+
p
?
)
, and the vector part of
p
is
1
/
2
(
p
?
p
?
)
.
The
square root
of the product of a quaternion with its conjugate is called its
norm
and is denoted
‖
q
‖
(Hamilton called this quantity the
tensor
of
q
, but this conflicts with the modern meaning of "
tensor
"). In formulas, 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
α
is real, then
This is a special case of the fact that the norm is
multiplicative
, meaning that
for any two quaternions
p
and
q
. Multiplicativity is a consequence of the formula for the conjugate of a product.
Alternatively it follows from the identity
(where
i
denotes the usual
imaginary unit
) and hence from the multiplicative property of
determinants
of square matrices.
This norm makes it possible to define the
distance
d
(
p
,
q
)
between
p
and
q
as the norm of their difference:
This makes
a
metric space
.
Addition and multiplication are
continuous
in regard to the associated
metric topology
.
This follows with exactly the same proof as for the real numbers
from the fact that
is a normed algebra.
Unit quaternion
[
edit
]
A
unit quaternion
is a quaternion of norm one. Dividing a nonzero quaternion
q
by its norm produces a unit quaternion
U
q
called the
versor
of
q
:
Every nonzero quaternion has a unique
polar decomposition
, while the zero quaternion can be formed from any unit quaternion.
Using conjugation and the norm makes it possible to define the
reciprocal
of a nonzero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of
and
is 1 (for either order of multiplication). So the
reciprocal
of
q
is defined to be
Since the multiplication is non-commutative, the quotient quantities
p?q
?1
or
q
?1
p
are different (except if
p
and
q
are scalar multiples of each other or if one is a scalar): the notation
p
/
q
is ambiguous and should not be used.
Algebraic properties
[
edit
]
The set
of all quaternions is a vector space over the real numbers with
dimension
4.
[c]
Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, 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 real numbers are also very rare:
Hurwitz's theorem
says that there are only four:
,
,
, and
(the octonions). 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
{±1, ±
i
, ±
j
, ±
k
}
forms a group under multiplication. This non-
abelian group
is called the quaternion group and is denoted
Q
8
.
[26]
The real
group ring
of
Q
8
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
1 + (?1)
,
i
+ (?
i
)
,
j
+ (?
j
)
, and
k
+ (?
k
)
. Here the first term in each of the differences is one of the basis elements
1,
i
,
j
, and
k
, and the second term is one of basis elements
?1, ?
i
, ?
j
, and
?
k
, not the additive inverses of
1,
i
,
j
, and
k
.
Quaternions and three-dimensional geometry
[
edit
]
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.
[16]
For the remainder of this section,
i
,
j
, and
k
will denote both the three imaginary
[27]
basis vectors of
and a basis for
Replacing
i
by
?
i
,
j
by
?
j
, and
k
by
?
k
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
p
=
b
1
i
+
c
1
j
+
d
1
k
and
q
=
b
2
i
+
c
2
j
+
d
2
k
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
pq
?
,
qp
?
,
p
?
q
, and
q
?
p
. Note that their vector parts are different.
The
cross product
of
p
and
q
relative to the orientation determined by the ordered basis
i
,
j
, and
k
is
(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the product
pq
(as quaternions), as well as the vector part of
?
q
?
p
?
. It also has the formula
For the
commutator
,
[
p
,
q
] =
pq
?
qp
, of two vector quaternions one obtains
In general, let
p
and
q
be quaternions and write
where
p
s
and
q
s
are the scalar parts, and
p
v
and
q
v
are the vector parts of
p
and
q
. 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
[28]
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.
[28]
Also see
Quaternions and spatial rotation
for more information about modeling three-dimensional rotations using quaternions.
See
Hanson
(2005)
[29]
for visualization of quaternions.
Matrix representations
[
edit
]
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. These are
injective
homomorphisms
from
to the
matrix rings
M(2,
C
)
and
M(4,
R
)
, respectively.
The quaternion
a
+
b
i
+
c
j
+
d
k
can be represented using a 2 × 2 complex matrix as
This representation has the following properties:
- Constraining any two of
b
,
c
and
d
to zero produces a representation of complex numbers. For example, setting
c
=
d
= 0
produces a diagonal complex matrix representation of complex numbers, and setting
b
=
d
= 0
produces a real matrix representation.
- The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the
determinant
of the corresponding matrix.
[30]
- The scalar part of a quaternion is one half of the
matrix trace
.
- The conjugate of a quaternion corresponds to the
conjugate transpose
of the matrix.
- By restriction this representation yields an
isomorphism
between the subgroup of unit quaternions and their image
SU(2)
. Topologically, the
unit quaternions
are the 3-sphere, so the underlying space of SU(2) is also a 3-sphere. The group
SU(2)
is important for describing
spin
in quantum mechanics; see
Pauli matrices
.
- There is a strong relation between quaternion units and Pauli matrices. The 2 × 2 complex matrix above can be written as
, so in this representation the quaternion units
{±1, ±
i
, ±
j
, ±
k
}
correspond to
{±
I
,
,
,
}
=
{±
I
,
,
,
}
, so multiplying any two Pauli matrices always yields a quaternion unit matrix, all of them except for ?1. One obtains ?1 via
i
2
=
j
2
=
k
2
=
i j k
= ?1
; e.g. the last equality is
- The representation in
M(2,
C
)
is not unique. A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose
- This gives an alternative representation,
[31]
- a
+
b
i
+
c
j
+
d
k
Using 4 × 4 real matrices, that same quaternion can be written as
However, the representation of quaternions in
M(4,
R
)
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
1,
i
,
j
, and
k
to the matrices in the quadruple is a homomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.
[32]
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. The scalar part of a quaternion is one quarter of the matrix trace. 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
c
=
d
= 0
.
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
×
|
a
|
d
|
?
b
|
?
c
|
a
|
a
|
d
|
?b
|
?c
|
?d
|
?d
|
a
|
c
|
?b
|
b
|
b
|
?
c
|
a
|
?
d
|
c
|
c
|
b
|
d
|
a
|
which is isomorphic ? through
? to
×
|
1
|
k
|
?
i
|
?
j
|
1
|
1
|
k
|
?
i
|
?
j
|
?
k
|
?
k
|
1
|
j
|
?
i
|
i
|
i
|
?
j
|
1
|
?
k
|
j
|
j
|
i
|
k
|
1
|
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 (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); 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
1
with
a
,
i
with
b
,
j
with
c
, and
k
with
d
and removing the row and column headers yields a matrix representation of
a
+
b
i
+
c
j
+
d
k
.
Lagrange's four-square theorem
[
edit
]
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
[
edit
]
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
1
and
j
. A vector in
can be written in terms of the basis elements
1
and
j
as
If we define
j
2
= ?1
and
i
j
= ?
j
i
, then we can multiply two vectors using the distributive law. Using
k
as an abbreviated notation for the product
i
j
leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion
a
+
b i
+
c
j
+
d
k
. If we write the elements of
as ordered pairs and quaternions as quadruples, then the correspondence is
Square roots
[
edit
]
Square roots of ?1
[
edit
]
In the complex numbers,
there are exactly two numbers,
i
and
?
i
, that give ?1 when squared. 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
q
=
a
+
b
i
+
c
j
+
d
k
be a quaternion, and assume that its square is ?1. In terms of
a
,
b
,
c
, and
d
, this means
To satisfy the last three equations, either
a
= 0
or
b
,
c
, and
d
are all 0. The latter is impossible because
a
is a real number and the first equation would imply that
a
2
= ?1
.
Therefore,
a
= 0
and
b
2
+
c
2
+
d
2
= 1
.
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 (or one in the case of 0).
[
citation needed
]
[d]
As a union of complex planes
[
edit
]
Each
antipodal pair
of square roots of ?1 creates a distinct copy of the complex numbers inside the quaternions. If
q
2
= ?1
,
then the copy is the
image
of the function
This is an
injective
ring homomorphism
from
to
which defines a field
isomorphism
from
onto its
image
. The images of the embeddings corresponding to
q
and ?
q
are identical.
Every non-real quaternion generates a
subalgebra
of the quaternions that is isomorphic to
and is thus a planar subspace of
write
q
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
:
(This is not the same as
.) The versor of the vector part of
q
,
, is a right versor with ?1 as its square. A straightforward verification shows that
defines an injective
homomorphism
of
normed algebras
from
into the quaternions. Under this homomorphism,
q
is the image of the complex number
.
As
is the
union
of the images of all these homomorphisms, one can view the quaternions as a
pencil of planes
intersecting on the
real line
. Each of these
complex planes
contains exactly one pair of
antipodal points
of the sphere of square roots of minus one.
Commutative subrings
[
edit
]
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
p
and
q
commute (i.e.,
p q
=
q p
) 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
.
Square roots of arbitrary quaternions
[
edit
]
Any quaternion
(represented here in scalar?vector representation) has at least one square root
which solves the equation
. Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions
where
is the norm of
and
is the norm of
. For any scalar quaternion
, this equation provides the correct square roots if
is interpreted as an arbitrary unit vector.
Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on
, i.e., where the scalar part is zero and the vector part is located on the
2-sphere
with radius
.
Functions of a quaternion variable
[
edit
]
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.
[36]
Exponential, logarithm, and power functions
[
edit
]
Given a quaternion,
the exponential is computed as
[37]
and the logarithm is
[37]
It follows that the polar decomposition of a quaternion may be written
where the
angle
[e]
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 (real) exponent
x
is given by:
Geodesic norm
[
edit
]
The
geodesic distance
d
g
(
p
,
q
)
between unit quaternions
p
and
q
is defined as:
[39]
and amounts to the absolute value of half the angle subtended by
p
and
q
along a
great arc
of the
S
3
sphere.
This angle can also be computed from the quaternion
dot product
without the logarithm as:
Three-dimensional and four-dimensional rotation groups
[
edit
]
The word "
conjugation
", besides the meaning given above, can also mean taking an element
a
to
r?a?r
?1
where
r
is some nonzero quaternion. All
elements that are conjugate to a given element
(in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)
[40]
Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of
consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part
cos(
φ
)
is a rotation by an angle
2
φ
, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:
[41]
The set of all unit quaternions (
versors
) forms a 3-sphere
S
3
and a group (a
Lie 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
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
SU(2)
, the group of complex
unitary
2×2 matrices of
determinant
1.
Let
A
be the set of quaternions of the form
a
+
b
i
+
c
j
+
d
k
where
a, b, c,
and
d
are either all
integers
or all
half-integers
. The set
A
is a ring (in fact a
domain
) 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
Schlafli symbol
{3,4,3}.
They correspond to the double cover of the rotational symmetry group of the regular
tetrahedron
. Similarly, the vertices of a
regular 600 cell
with Schlafli symbol
{3,3,5
} 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
.
[42]
Quaternion algebras
[
edit
]
The Quaternions can be generalized into further algebras called
quaternion algebras
. Take
F
to be any field with characteristic different from 2, and
a
and
b
to be elements of
F
; a four-dimensional unitary associative algebra can be defined over
F
with basis
1,
i
,
j
,
and
i j
, where
i
2
=
a
,
j
2
=
b
and
i j
= ?
j i
(so
(i j)
2
= ?
a b
).
Quaternion algebras 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
.
Quaternions as the even part of
Cl
3,0
(R)
[
edit
]
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
σ
1
,
σ
2
,
σ
3
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
r
in a plane perpendicular to a unit vector
w
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
σ
1
and
σ
2
. This is very similar to the corresponding quaternion formula,
Indeed, the two structures
and
are
isomorphic
. One natural identification is
and it is straightforward to confirm that this preserves the Hamilton relations
In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but to
bivectors
? quantities with
magnitudes
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
σ
1
and
σ
2
, there is only one bivector basis element
σ
1
σ
2
, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elements
σ
2
σ
3
,
σ
3
σ
1
,
σ
1
σ
2
, 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:
[43]
- Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
- In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
- Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
- In the
conformal model
of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
- Rotor-encoded transformations make interpolation particularly straightforward.
- Rotors carry over naturally to
pseudo-Euclidean spaces
, for example, the
Minkowski space
of
special relativity
. In such spaces rotors can be used to efficiently represent
Lorentz boosts
, and to interpret formulas involving the
gamma matrices
.
For further detail about the geometrical uses of Clifford algebras, see
Geometric algebra
.
Brauer group
[
edit
]
The quaternions are "essentially" the only (non-trivial)
central simple algebra
(CSA) over the real numbers, in the sense that every CSA over the real numbers is
Brauer equivalent
to either the real numbers or the quaternions. Explicitly, the
Brauer group
of the real numbers consists of two classes, represented by the real numbers 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
(specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.
CSAs ? finite dimensional rings over a field, which are
simple algebras
(have no non-trivial 2-sided ideals, just as with fields) 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 real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.
Quotations
[
edit
]
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 (circa 1848)
[44]
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 (circa 1853)
[45]
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
.
There was a time, indeed, when I, although recognizing the appropriateness of vector analysis in electromagnetic theory (and in mathematical physics generally), did think it was harder to understand and to work than the Cartesian analysis. But that was before I had thrown off the quaternionic old-man-of-the-sea who fastened himself about my shoulders when reading the only accessible treatise on the subject ? Prof. Tait's Quaternions. But I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work. There is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). The vector analysis I use may be described either as a convenient and systematic abbreviation of Cartesian analysis; or else, as Quaternions without the quaternions, .... "Quaternion" was, I think, defined by an American schoolgirl to be "an ancient religious ceremony". This was, however, a complete mistake. The ancients ? unlike Prof. Tait ? knew not, and did not worship Quaternions.
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 everyday 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.
... 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)
[49]
See also
[
edit
]
Notes
[
edit
]
- ^
A more personal view of quaternions was written by
Joachim Lambek
in 1995. He wrote in his essay
If Hamilton had prevailed: quaternions in physics
: "My own interest as a graduate student was raised by the inspiring book by Silberstein". He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics."
Lambek, J. (1995). "If Hamilton had prevailed: Quaternions in physics".
Math. Intelligencer
. Vol. 17, no. 4. pp. 7?15.
doi
:
10.1007/BF03024783
.
- ^
It is important to note that the vector part of a quaternion is, in truth, an "axial" vector or "
pseudovector
",
not
an ordinary or "polar" vector, as was formally proven by Altmann (1986).
[23]
A polar vector can be represented in calculations (for example, for rotation by a quaternion "similarity transform") by a pure imaginary quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.
- ^
In comparison, the real numbers
have dimension 1, the complex numbers
have dimension 2, and the octonions
have dimension 8.
- ^
The identification of the square roots of minus one in
was given by Hamilton
[33]
but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his three-page exposition included in
Historical Topics in Algebra
published by the
National Council of Teachers of Mathematics
.
[34]
More recently, the sphere of square roots of minus one is described in
Ian R. Porteous
's book
Clifford Algebras and the Classical Groups
(Cambridge, 1995) in proposition 8.13.
[35]
- ^
Books on applied mathematics, such as Corke (2017)
[38]
often use different notation with
φ
:=
1
/
2
θ
? that is,
another variable
θ
= 2
φ
.
References
[
edit
]
- ^
"On Quaternions; or on a new System of Imaginaries in Algebra".
Letter to John T. Graves
. 17 October 1843.
- ^
Rozenfel?d, Boris Abramovich (1988).
The history of non-euclidean geometry: Evolution of the concept of a geometric space
. Springer. p. 385.
ISBN
9780387964584
.
- ^
Hamilton
. Hodges and Smith. 1853. p.
60
.
quaternion quotient lines tridimensional space time
- ^
Hardy 1881
. Ginn, Heath, & co. 1881. p. 32.
ISBN
9781429701860
.
- ^
Curtis, Morton L. (1984),
Matrix Groups
(2nd ed.), New York:
Springer-Verlag
, p. 10,
ISBN
978-0-387-96074-6
- ^
a
b
Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010).
"Orientational Sampling Schemes Based on Four Dimensional Polytopes"
.
Symmetry
.
2
(3): 1423?1449.
Bibcode
:
2010Symm....2.1423M
.
doi
:
10.3390/sym2031423
.
- ^
Kunze, Karsten; Schaeben, Helmut (November 2004). "The Bingham distribution of quaternions and its spherical radon transform in texture analysis".
Mathematical Geology
.
36
(8): 917?943.
Bibcode
:
2004MatGe..36..917K
.
doi
:
10.1023/B:MATG.0000048799.56445.59
.
S2CID
55009081
.
- ^
Smith, Frank (Tony).
"Why not sedenion?"
. Retrieved
8 June
2018
.
- ^
a
b
c
See
Hazewinkel, Gubareni & Kirichenko 2004
, p.
12
- ^
Conway & Smith 2003
, p.
9
- ^
Bradley, Robert E.; Sandifer, Charles Edward (2007).
Leonhard Euler: life, work and legacy
. Elsevier. p. 193.
ISBN
978-0-444-52728-8
.
They mention
Wilhelm Blaschke
's claim in 1959 that "the quaternions were first identified by L. Euler in a letter to Goldbach written on 4 May 1748," and they comment that "it makes no sense whatsoever to say that Euler "identified" the quaternions in this letter ... this claim is absurd."
- ^
Pujol, J., "
Hamilton, Rodrigues, Gauss, Quaternions, and Rotations: A Historical Reassessment
"
Communications in Mathematical Analysis
(2012), 13(2), 1?14
- ^
Gauss, C.F. (1900). "Mutationen des Raumes [Transformations of space] (c. 1819)". In Martin Brendel (ed.).
Carl Friedrich Gauss Werke
[
The works of Carl Friedrich Gauss
]. Vol. 8. article edited by Prof. Stackel of Kiel, Germany. Gottingen, DE: Koniglichen Gesellschaft der Wissenschaften [Royal Society of Sciences]. pp. 357?361.
- ^
a
b
Hamilton, W.R. (1844). "Letter".
London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
. Vol. xxv. pp. 489?495.
- ^
Hamilton, Sir W.R.
(1866).
Hamilton, W.E.
(ed.).
Elements of Quaternions
. London: Longmans, Green, & Co.
- ^
a
b
Shoemake, Ken
(1985).
"Animating Rotation with Quaternion Curves"
(PDF)
.
Computer Graphics
.
19
(3): 245?254.
doi
:
10.1145/325165.325242
.
Presented at
SIGGRAPH
'85.
- ^
Tomb Raider
(1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth three-dimensional rotations. See, for example
Nick Bobick (July 1998).
"Rotating objects using quaternions"
.
Game Developer
.
- ^
McCarthy, J.M. (1990).
An Introduction to Theoretical Kinematics
. MIT Press.
ISBN
978-0-262-13252-7
.
- ^
Hurwitz, A. (1919),
Vorlesungen uber die Zahlentheorie der Quaternionen
, Berlin: J. Springer,
JFM
47.0106.01
, concerning
Hurwitz quaternions
- ^
Huerta, John (27 September 2010).
"Introducing The Quaternions"
(PDF)
.
Archived
(PDF)
from the original on 2014-10-21
. Retrieved
8 June
2018
.
- ^
Wood, Charlie (6 September 2018).
"The Strange Numbers That Birthed Modern Algebra"
.
Abstractions blog
. Quanta Magazine.
- ^
Eves (1976
, p. 391)
- ^
Altmann, S.L.
Rotations, Quaternions, and Double Groups
. Ch. 12.
- ^
Hamilton, Sir William Rowan (1866). "Article 285".
Elements of Quaternions
. Longmans, Green, & Company. p.
310
.
- ^
Hardy (1881).
"Elements of Quaternions"
.
Science
.
2
(75). library.cornell.edu: 65.
doi
:
10.1126/science.os-2.75.564
.
PMID
17819877
.
- ^
"quaternion group"
.
Wolframalpha.com
.
- ^
Gibbs, J. Willard; Wilson, Edwin Bidwell (1901).
Vector Analysis
. Yale University Press. p.
428
.
right tensor dyadic
- ^
a
b
Hamilton, W.R.
(1844?1850).
"On quaternions or a new system of imaginaries in algebra"
. David R. Wilkins collection.
Philosophical Magazine
.
Trinity College Dublin
.
- ^
"Visualizing Quaternions"
. Morgan-Kaufmann/Elsevier. 2005.
- ^
"[no title cited; determinant evaluation]"
.
Wolframalpha.com
.
- ^
eg Altmann (1986),
Rotations, Quaternions, and Double Groups
, p. 212, eqn 5
- ^
Farebrother, Richard William; Groß, Jurgen; Troschke, Sven-Oliver (2003).
"Matrix representation of quaternions"
.
Linear Algebra and Its Applications
.
362
: 251?255.
doi
:
10.1016/s0024-3795(02)00535-9
.
- ^
Hamilton, W.R. (1899).
Elements of Quaternions
(2nd ed.). Cambridge University Press. p. 244.
ISBN
1-108-00171-8
.
- ^
Perlis, Sam (1971).
"Capsule 77: Quaternions"
.
Historical Topics in Algebra
. Historical Topics for the Mathematical Classroom. Vol. 31. Reston, VA:
National Council of Teachers of Mathematics
. p. 39.
ISBN
9780873530583
.
OCLC
195566
.
- ^
Porteous, Ian R.
(1995). "Chapter 8: Quaternions".
Clifford Algebras and the Classical Groups
(PDF)
. Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge:
Cambridge University Press
. p. 60.
doi
:
10.1017/CBO9780511470912.009
.
ISBN
9780521551779
.
MR
1369094
.
OCLC
32348823
.
- ^
"[no title cited]"
(PDF)
.
bridgesmathart.org
. archive
. Retrieved
19 August
2018
.
- ^
a
b
Sarkka, Simo (June 28, 2007).
"Notes on Quaternions"
(PDF)
.
Lce.hut.fi
. Archived from
the original
(PDF)
on 5 July 2017.
- ^
Corke, Peter (2017).
Robotics, Vision, and Control ? Fundamental Algorithms in MATLAB
.
Springer
.
ISBN
978-3-319-54413-7
.
- ^
Park, F.C.; Ravani, Bahram (1997).
"Smooth invariant interpolation of rotations"
.
ACM Transactions on Graphics
.
16
(3): 277?295.
doi
:
10.1145/256157.256160
.
S2CID
6192031
.
- ^
Hanson, Jason (2011). "Rotations in three, four, and five dimensions".
arXiv
:
1103.5263
[
math.MG
].
- ^
Guna?ti, Gokmen (2016).
Quaternions Algebra, Their Applications in Rotations and Beyond Quaternions
(BS). Linnaeus University.
- ^
"Three-Dimensional Point Groups"
.
www.classe.cornell.edu
. Retrieved
2022-12-09
.
- ^
"Quaternions and Geometric Algebra"
.
geometricalgebra.net
. Retrieved
2008-09-12
.
See also:
Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007).
Geometric Algebra for Computer Science
.
Morgan Kaufmann
.
ISBN
978-0-12-369465-2
.
- ^
Hamilton, William Rowan (1853).
Lectures on quaternions
. Dublin: Hodges and Smith. p. 522.
- ^
Graves, R.P.
Life of Sir William Rowan Hamilton
. Dublin Hodges, Figgis. pp. 635?636.
- ^
Thompson, Silvanus Phillips (1910).
The life of William Thomson (Vol. 2)
. London, Macmillan. p. 1138.
- ^
Heaviside, Oliver
(1893).
Electromagnetic Theory
. Vol. I. London, UK: The Electrician Printing and Publishing Company. pp. 134?135.
- ^
Ludwik Silberstein (1924).
Preface to second edition
of
The Theory of Relativity
- ^
Altmann, Simon L. (1986).
Rotations, quaternions, and double groups
. Clarendon Press.
ISBN
0-19-855372-2
.
LCCN
85013615
.
Further reading
[
edit
]
Books and publications
[
edit
]
- Hamilton, William Rowan
(1844).
"On quaternions, or on a new system of imaginaries in algebra"
.
Philosophical Magazine
.
25
(3): 489?495.
doi
:
10.1080/14786444408645047
.
- Hamilton, William Rowan
(1853), "
Lectures on Quaternions
". Royal Irish Academy.
- Hamilton (1866)
Elements of Quaternions
University of Dublin
Press. Edited by William Edwin Hamilton, son of the deceased author.
- Hamilton (1899)
Elements of Quaternions
volume I, (1901) volume II. Edited by
Charles Jasper Joly
; published by
Longmans, Green & Co.
- 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), "
"Archived copy"
. Archived from the original on August 8, 2014
. Retrieved
June 26,
2005
.
{{
cite web
}}
: CS1 maint: archived copy as title (
link
) CS1 maint: unfit URL (
link
)
". M.A. Sec. R.S.E.
Encyclopædia Britannica
, Ninth Edition, 1886, Vol. XX, pp. 160–164. (bzipped
PostScript
file)
- Joly, Charles Jasper (1905).
A manual of quaternions
. Macmillan.
LCCN
05036137
.
- Macfarlane, Alexander
(1906).
Vector analysis and quaternions
(4th ed.). Wiley.
LCCN
16000048
.
- Chisholm, Hugh
, ed. (1911).
"Algebra"
.
Encyclopædia Britannica
(11th ed.). Cambridge University Press.
(
See section on quaternions.
)
- Finkelstein, David; Jauch, Josef M.; Schiminovich, Samuel; Speiser, David (1962).
"Foundations of quaternion quantum mechanics"
.
J. Math. Phys
.
3
(2): 207?220.
Bibcode
:
1962JMP.....3..207F
.
doi
:
10.1063/1.1703794
.
S2CID
121453456
.
- Du Val, Patrick
(1964).
Homographies, quaternions, and rotations
. Oxford mathematical monographs. Clarendon Press.
LCCN
64056979
.
- Michael J. Crowe (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, Mobius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
- Altmann, Simon L. (1989). "Hamilton, Rodrigues, and the Quaternion Scandal".
Mathematics Magazine
.
62
(5): 291?308.
doi
:
10.1080/0025570X.1989.11977459
.
- Pujol, Jose (2014). "On Hamilton's Nearly-Forgotten Early Work on the Relation between Rotations and Quaternions and on the Composition of Rotations".
The American Mathematical Monthly
.
121
(6): 515?522.
doi
:
10.4169/amer.math.monthly.121.06.515
.
S2CID
1543951
.
- Adler, Stephen L. (1995).
Quaternionic quantum mechanics and quantum fields
. International series of monographs on physics. Vol. 88. Oxford University Press.
ISBN
0-19-506643-X
.
LCCN
94006306
.
- Ward, J.P. (1997).
Quaternions and Cayley Numbers: Algebra and Applications
. Kluwer Academic.
ISBN
0-7923-4513-4
.
- Kantor, I.L.; Solodnikov, A.S. (1989).
Hypercomplex numbers, an elementary introduction to algebras
. Springer-Verlag.
ISBN
0-387-96980-2
.
- Gurlebeck, Klaus; Sprossig, Wolfgang (1997).
Quaternionic and Clifford calculus for physicists and engineers
. Mathematical methods in practice. Vol. 1. Wiley.
ISBN
0-471-96200-7
.
LCCN
98169958
.
- Kuipers, Jack (2002).
Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality
.
Princeton University Press
.
ISBN
0-691-10298-8
.
- Conway, John Horton
; Smith, Derek A. (2003).
On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry
. A.K. Peters.
ISBN
1-56881-134-9
.
(
review
).
- Jack, P.M. (2003). "Physical space as a quaternion structure, I: Maxwell equations. A brief Note".
arXiv
:
math-ph/0307038
.
- Kravchenko, Vladislav
(2003).
Applied Quaternionic Analysis
. Heldermann Verlag.
ISBN
3-88538-228-8
.
- Hazewinkel, Michiel
; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004).
Algebras, rings and modules
. Vol. 1. Springer.
ISBN
1-4020-2690-0
.
- Hanson, Andrew J. (2006).
Visualizing Quaternions
. Elsevier.
ISBN
0-12-088400-3
.
- Binz, Ernst; Pods, Sonja (2008). "1. The Skew Field of Quaternions".
Geometry of Heisenberg Groups
.
American Mathematical Society
.
ISBN
978-0-8218-4495-3
.
- Doran, Chris J.L.
; Lasenby, Anthony N. (2003).
Geometric Algebra for Physicists
. Cambridge University Press.
ISBN
978-0-521-48022-2
.
- Vince, John A. (2008).
Geometric Algebra for Computer Graphics
. Springer.
ISBN
978-1-84628-996-5
.
- For molecules that can be regarded as classical rigid bodies
molecular dynamics
computer simulation employs quaternions. They were first introduced for this purpose by
Evans, D.J. (1977). "On the Representation of Orientation Space".
Mol. Phys
.
34
(2): 317?325.
Bibcode
:
1977MolPh..34..317E
.
doi
:
10.1080/00268977700101751
.
- Zhang, Fuzhen (1997).
"Quaternions and Matrices of Quaternions"
.
Linear Algebra and Its Applications
.
251
: 21?57.
doi
:
10.1016/0024-3795(95)00543-9
.
- Ron Goldman (2010).
Rethinking Quaternions: Theory and Computation
. Morgan & Claypool.
ISBN
978-1-60845-420-4
.
- Eves, Howard (1976),
An Introduction to the History of Mathematics
(4th ed.), New York:
Holt, Rinehart and Winston
,
ISBN
0-03-089539-1
- Voight, John (2021).
Quaternion Algebras
. Graduate Texts in Mathematics. Vol. 288. Springer.
doi
:
10.1007/978-3-030-56694-4
.
ISBN
978-3-030-57467-3
.
Links and monographs
[
edit
]
- "Quaternion Notices"
.
Notices and materials related to Quaternion conference presentations
- "Quaternion"
,
Encyclopedia of Mathematics
,
EMS Press
, 2001 [1994]
- "Frequently Asked Questions"
.
Matrix and Quaternion
. 1.21.
- Sweetser, Doug.
"Doing Physics with Quaternions"
.
- Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)
- Gsponer, Andre; Hurni, Jean-Pierre (2002). "The Physical Heritage of Sir W. R. Hamilton".
arXiv
:
math-ph/0201058
.
- Wilkins, D.R.
"Hamilton's Research on Quaternions"
.
- Grossman, David J.
"Quaternion Julia Fractals"
.
3D Raytraced Quaternion
Julia Fractals
- "Quaternion Math and Conversions"
.
Great page explaining basic math with links to straight forward rotation conversion formulae.
- Mathews, John H.
"Bibliography for Quaternions"
. Archived from
the original
on 2006-09-02.
- "Quaternion powers"
. GameDev.net.
- Hanson, Andrew.
"Visualizing Quaternions home page"
. Archived from
the original
on 2006-11-05.
- Karney, Charles F.F. (January 2007). "Quaternions in molecular modeling".
J. Mol. Graph. Mod
.
25
(5): 595?604.
arXiv
:
physics/0506177
.
doi
:
10.1016/j.jmgm.2006.04.002
.
PMID
16777449
.
S2CID
6690718
.
- Mebius, Johan E. (2005). "A matrix-based proof of the quaternion representation theorem for four-dimensional rotations".
arXiv
:
math/0501249
.
- Mebius, Johan E. (2007). "Derivation of the Euler?Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations".
arXiv
:
math/0701759
.
- "Hamilton Walk"
. Department of Mathematics,
NUI Maynooth
.
- "Using Quaternions to represent rotation"
.
OpenGL:Tutorials
. Archived from
the original
on 2007-12-15.
- David Erickson,
Defence Research and Development Canada
(DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper.
- Martinez, Alberto.
"Negative Math, How Mathematical Rules Can Be Positively Bent"
. Department of History, University of Texas. Archived from
the original
on 2011-09-24.
- Stahlke, D.
"Quaternions in Classical Mechanics"
(PDF)
.
- Morier-Genoud, Sophie; Ovsienko, Valentin (2008). "Well, Papa, can you multiply triplets?".
arXiv
:
0810.5562
[
math.AC
].
describes how the quaternions can be made into a skew-commutative algebra graded by
Z
/2 ×
Z
/2 ×
Z
/2
.
- Joyce, Helen (November 2004).
"Curious Quaternions"
. hosted by
John Baez
.
- Ibanez, Luis.
"Tutorial on Quaternions. Part I"
(PDF)
. Archived from
the original
(PDF)
on 2012-02-04
. Retrieved
2011-12-05
.
Part II
(PDF; using Hamilton's terminology, which differs from the modern usage)
- Ghiloni, R.; Moretti, V.; Perotti, A. (2013). "Continuous slice functional calculus in quaternionic Hilbert spaces".
Rev. Math. Phys
.
25
(4): 1350006?126.
arXiv
:
1207.0666
.
Bibcode
:
2013RvMaP..2550006G
.
doi
:
10.1142/S0129055X13500062
.
S2CID
119651315
.
Ghiloni, R.; Moretti, V.; Perotti, A. (2017). "Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures".
Rev. Math. Phys
.
29
: 1750034.
arXiv
:
1602.02661
.
doi
:
10.1142/S0129055X17500349
.
S2CID
124709652
.
two expository papers about continuous functional calculus and spectral theory in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
- Quaternions
the Android app shows the quaternion corresponding to the orientation of the device.
- Rotating Objects Using Quaternions
article speaking to the use of Quaternions for rotation in video games/computer graphics.
External links
[
edit
]
Look up
quaternion
in Wiktionary, the free dictionary.