In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of rotations about a fixed point in
four-dimensional Euclidean space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
is denoted SO(4). The name comes from the fact that it is the
special orthogonal group of order 4.
In this article ''
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
'' means ''rotational displacement''. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise.
A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.
Geometry of 4D rotations
Four-dimensional rotations are of two types: simple rotations and double rotations.
Simple rotations
A simple rotation about a rotation centre leaves an entire plane through (axis-plane) fixed. Every plane that is completely
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to intersects in a certain point . Each such point is the centre of the 2D rotation induced by in . All these 2D rotations have the same rotation angle .
Half-line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s from in the axis-plane are not displaced; half-lines from orthogonal to are displaced through ; all other half-lines are displaced through an angle less than .
Double rotations
For each rotation of 4-space (fixing the origin), there is at least one pair of
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
2-planes and each of which is invariant and whose direct sum is all of 4-space. Hence operating on either of these planes produces an ordinary rotation of that plane. For almost all (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles in plane and in plane – both assumed to be nonzero – are different. The unequal rotation angles and satisfying , are almost uniquely determined by . Assuming that 4-space is oriented, then the orientations of the 2-planes and can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (), is sometimes termed a "double rotation".
In that case of a double rotation, and are the only pair of invariant planes, and
half-line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s from the origin in , are displaced through and respectively, and half-lines from the origin not in or are displaced through angles strictly between and .
Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
planes instead of just two, and all
half-line
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s from are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.
Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation , and take an orientation-consistent ordered set of mutually perpendicular half-lines at (denoted as ) such that and span an invariant plane, and therefore and also span an invariant plane. Now assume that only the rotation angle is specified. Then there are in general four isoclinic rotations in planes and with rotation angle , depending on the rotation senses in and .
We make the convention that the rotation senses from to and from to are reckoned positive. Then we have the four rotations , , and . and are each other's
inverses; so are and . As long as lies between 0 and , these four rotations will be distinct.
Isoclinic rotations with like signs are denoted as ''left-isoclinic''; those with opposite signs as ''right-isoclinic''. Left- and right-isoclinic rotations are represented respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below.
The four rotations are pairwise different except if or . The angle corresponds to the identity rotation; corresponds to the
central inversion
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic.
Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However, when another isoclinic rotation with its own axes , , , is selected, then one can always choose the
order of , , , such that can be transformed into by a rotation rather than by a rotation-reflection (that is, so that the ordered basis , , , is also consistent with the same fixed choice of orientation as , , , ). Therefore, once one has selected an orientation (that is, a system of axes that is universally denoted as right-handed), one can determine the left or right character of a specific isoclinic rotation.
Group structure of SO(4)
SO(4) is a
noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
6-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
.
Each plane through the rotation centre is the axis-plane of a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to SO(2). All these subgroups are mutually
conjugate in SO(4).
Each pair of completely
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
planes through is the pair of
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
planes of a commutative subgroup of SO(4) isomorphic to .
These groups are
maximal tori of SO(4), which are all mutually conjugate in SO(4). See also
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...
.
All left-isoclinic rotations form a noncommutative subgroup of SO(4), which is isomorphic to the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of unit
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s. All right-isoclinic rotations likewise form a subgroup of SO(4) isomorphic to . Both and are maximal subgroups of SO(4).
Each left-isoclinic rotation
commutes with each right-isoclinic rotation. This implies that there exists a
direct product with
normal subgroups and ; both of the corresponding
factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to . (This is not SO(4) or a subgroup of it, because and are not disjoint: the identity and the central inversion each belong to both and .)
Each 4D rotation is in two ways the product of left- and right-isoclinic rotations and . and are together determined up to the central inversion, i.e. when both and are multiplied by the central inversion their product is again.
This implies that is the
universal covering group of SO(4) — its unique
double cover — and that and are normal subgroups of SO(4). The identity rotation and the central inversion form a group of order 2, which is the
centre
Center or centre may refer to:
Mathematics
* Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentri ...
of SO(4) and of both and . The centre of a group is a normal subgroup of that group. The factor group of C
2 in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of
3L by C
2 and of
3R by C
2 are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by
3L and of SO(4) by
3R are each isomorphic to SO(3).
The topology of SO(4) is the same as that of the Lie group , namely the space
where
is the
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
of dimension 3 and
is the
3-sphere. However, it is noteworthy that, as a Lie group, SO(4) is not a direct product of Lie groups, and so it is not isomorphic to .
Special property of SO(4) among rotation groups in general
The odd-dimensional rotation groups do not contain the central inversion and are
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
s.
The even-dimensional rotation groups do contain the central inversion and have the group as their
centre
Center or centre may refer to:
Mathematics
* Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentri ...
. For even n ≥ 6, SO(n) is almost simple in that the
factor group SO(n)/C
2 of SO(n) by its centre is a simple group.
SO(4) is different: there is no
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
by any element of SO(4) that transforms left- and right-isoclinic rotations into each other.
Reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
s transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of ''all'' isometries with fixed point the distinct subgroups and are conjugate to each other, and so cannot be normal subgroups of O(4). The 5D rotation group SO(5) and all higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any two isoclinic rotations through the same angle are conjugate. The set of all isoclinic rotations is not even a subgroup of SO(2), let alone a normal subgroup.
Algebra of 4D rotations
SO(4) is commonly identified with the group of
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
-preserving
isometric linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
mappings of a 4D
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
with
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s onto itself.
With respect to an
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
in such a space SO(4) is represented as the group of real 4th-order
orthogonal matrices
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
+1.
Isoclinic decomposition
A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows:
Let
:
be its matrix with respect to an arbitrary orthonormal basis.
Calculate from this the so-called ''associate matrix''
:
has
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
one and is of unit
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
as a 16D vector if and only if is indeed a 4D rotation matrix. In this case there exist real numbers and such that
:
and
:
There are exactly two sets of and such that and . They are each other's opposites.
The rotation matrix then equals
:
This formula is due to Van Elfrinkhof (1897).
The first factor in this decomposition represents a left-isoclinic rotation, the second factor a right-isoclinic rotation. The factors are determined up to the negative 4th-order
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, i.e. the central inversion.
Relation to quaternions
A point in 4-dimensional space with
Cartesian coordinates may be represented by a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
.
A left-isoclinic rotation is represented by left-multiplication by a unit quaternion . In matrix-vector language this is
:
Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion , which is in matrix-vector form
:
In the preceding section (
#Isoclinic decomposition) it is shown how a general 4D rotation is split into left- and right-isoclinic factors.
In quaternion language Van Elfrinkhof's formula reads
:
or, in symbolic form,
:
According to the German mathematician
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
this formula was already known to Cayley in 1854.
Quaternion multiplication is
associative. Therefore,
:
which shows that left-isoclinic and right-isoclinic rotations commute.
The eigenvalues of 4D rotation matrices
The four
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of a 4D rotation matrix generally occur as two conjugate pairs of
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of unit magnitude. If an eigenvalue is real, it must be ±1, since a rotation leaves the magnitude of a vector unchanged. The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions and are equal in magnitude and have the same sign. If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of and are not equal then all eigenvalues are complex, and the rotation is a double rotation.
The Euler–Rodrigues formula for 3D rotations
Our ordinary 3D space is conveniently treated as the subspace with coordinate system 0XYZ of the 4D space with coordinate system UXYZ. Its
rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a ...
is identified with the subgroup of SO(4) consisting of the matrices
:
In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to , , , , or in quaternion representation: .
The 3D rotation matrix then becomes the
Euler–Rodrigues formula In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.
The rotation is described by four Euler para ...
for 3D rotations
:
which is the representation of the 3D rotation by its
Euler–Rodrigues parameters: .
The corresponding quaternion formula , where , or, in expanded form:
:
is known as the
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
–
Cayley formula.
Hopf coordinates
Rotations in 3D space are made mathematically much more tractable by the use of
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
. Any rotation in 3D can be characterized by a fixed axis of rotation and an invariant plane perpendicular to that axis. Without loss of generality, we can take the -plane as the invariant plane and the -axis as the fixed axis. Since radial distances are not affected by rotation, we can characterize a rotation by its effect on the unit sphere (2-sphere) by
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
referred to the fixed axis and invariant plane:
:
Because , the points lie on the 2-sphere. A point at rotated by an angle about the -axis is specified simply by . While
hyperspherical coordinates are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by
Hopf coordinates ,
[
] which are a set of three angular coordinates specifying a position on the 3-sphere. For example:
:
Because , the points lie on the 3-sphere.
In 4D space, every rotation about the origin has two invariant planes which are completely orthogonal to each other and intersect at the origin, and are rotated by two independent angles and . Without loss of generality, we can choose, respectively, the - and -planes as these invariant planes. A rotation in 4D of a point through angles and is then simply expressed in Hopf coordinates as .
Visualization of 4D rotations
Every rotation in 3D space has a fixed axis unchanged by rotation. The rotation is completely specified by specifying the axis of rotation and the angle of rotation about that axis. Without loss of generality, this axis may be chosen as the -axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation.
In 3D space, the
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
may be seen as a parametric expression of the 2-sphere. For fixed they describe circles on the 2-sphere which are perpendicular to the -axis and these circles may be viewed as trajectories of a point on the sphere. A point on the sphere, under a rotation about the -axis, will follow a trajectory as the angle varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: , with being an "angular velocity".
Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a point). The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the - and -planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation.
In 4D space, the Hopf angles parameterize the 3-sphere. For fixed they describe a torus parameterized by and , with being the special case of the
Clifford torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...
in the - and -planes. These tori are not the usual tori found in 3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be
stereographically projected onto the whole Euclidean 3D-space, and these tori are then seen as the usual tori of revolution. It can be seen that a point specified by undergoing a rotation with the - and -planes invariant will remain on the torus specified by .
The trajectory of a point can be written as a function of time as and stereographically projected onto its associated torus, as in the figures below.
In these figures, the initial point is taken to be , i.e. on the Clifford torus. In Fig. 1, two simple rotation trajectories are shown in black, while a left and a right isoclinic trajectory is shown in red and blue respectively. In Fig. 2, a general rotation in which and is shown, while in Fig. 3, a general rotation in which and is shown.
Generating 4D rotation matrices
Four-dimensional rotations can be derived from
Rodrigues' rotation formula
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform al ...
and the Cayley formula. Let be a 4 × 4
skew-symmetric matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
. The skew-symmetric matrix can be uniquely decomposed as
:
into two skew-symmetric matrices and satisfying the properties , and , where and are the eigenvalues of . Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices and by Rodrigues' rotation formula and the Cayley formula.
Let be a 4 × 4 nonzero skew-symmetric matrix with the set of eigenvalues
:
Then can be decomposed as
:
where and are skew-symmetric matrices satisfying the properties
:
Moreover, the skew-symmetric matrices and are uniquely obtained as
:
and
:
Then,
:
is a rotation matrix in , which is generated by Rodrigues' rotation formula, with the set of eigenvalues
:
Also,
:
is a rotation matrix in , which is generated by Cayley's rotation formula, such that the set of eigenvalues of is,
:
The generating rotation matrix can be classified with respect to the values and as follows:
# If and or vice versa, then the formulae generate simple rotations;
# If and are nonzero and , then the formulae generate double rotations;
# If and are nonzero and , then the formulae generate isoclinic rotations.
See also
*
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For t ...
*
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
*
Orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
*
Orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
*
Plane of rotation
In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as ...
*
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
*
Quaternions and spatial rotation Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation abou ...
Notes
References
Bibliography
*L. van Elfrinkhof
Eene eigenschap van de orthogonale substitutie van de vierde orde.''Handelingen van het 6e Nederlandsch Natuurkundig en Geneeskundig Congres, Delft, 1897.
*
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
: Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by E.R. Hedrick and C.A. Noble. The Macmillan Company, New York, 1932.
Henry Parker Manning ''Geometry of four dimensions''. The Macmillan Company, 1914. Republished unaltered and unabridged by Dover Publications in 1954. In this monograph four-dimensional geometry is developed from first principles in a synthetic axiomatic way. Manning's work can be considered as a direct extension of the works of
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
and
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
to four dimensions.
*J. H. Conway and D. A. Smith: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, 2003.
*
*
*
P.H.Schoute ''Mehrdimensionale Geometrie''. Leipzig: G.J.Göschensche Verlagshandlung. Volume 1 (Sammlung Schubert XXXV): Die linearen Räume, 1902. Volume 2 (Sammlung Schubert XXXVI): Die Polytope, 1905.
*
*
*
*
*{{cite journal , arxiv=2003.09236 , date=8 Jan 2021 , last=Zamboj , first=Michal , title=Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space , journal=Journal of Computational Design and Engineering , volume=8 , issue=3 , pages=836–854 , doi=10.1093/jcde/qwab018
Four-dimensional geometry
Quaternions
Rotation