SO(4)

In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4.

In this article ''rotation'' 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 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-lines 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 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-lines 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 planes instead of just two, and all half-lines 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, 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 compact 6-dimensional Lie group.

Each plane through the rotation centre is the axis-plane of a commutative subgroup isomorphic to SO(2). All these subgroups are mutually conjugate in SO(4).

Each pair of completely orthogonal planes through is the pair of invariant 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.

All left-isoclinic rotations form a noncommutative subgroup of SO(4), which is isomorphic to the multiplicative group of unit quaternions. 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 of SO(4) and of both and . The centre of a group is a normal subgroup of that group. The factor group of C₂ in SO(4) is isomorphic to SO(3) × SO(3). The factor groups of ³_L by C₂ and of ³_R by C₂ are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by ³_L and of SO(4) by ³_R are each isomorphic to SO(3).

The topology of SO(4) is the same as that of the Lie group , namely the space $\mathbb^3 \times \mathbb^3$ where $\mathbb^3$ is the real projective space of dimension 3 and $\mathbb^3$ 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 groups.

The even-dimensional rotation groups do contain the central inversion and have the group as their centre. For even n ≥ 6, SO(n) is almost simple in that the factor group SO(n)/C₂ of SO(n) by its centre is a simple group.

SO(4) is different: there is no conjugation by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. Reflections 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-preserving isometric linear mappings of a 4D vector space with inner product over the real numbers onto itself.

With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order orthogonal matrices with determinant +1.

Isoclinic decomposition

A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows:

Let
:$A= \begin a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ \\ \end$
be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called ''associate matrix''
:$M= \frac \begin a_+a_+a_+a_ & +a_-a_-a_+a_ & +a_+a_-a_-a_ & +a_-a_+a_-a_ \\ a_-a_+a_-a_ & -a_-a_+a_+a_ & +a_-a_-a_+a_ & -a_-a_-a_-a_ \\ a_-a_-a_+a_ & -a_-a_-a_-a_ & -a_+a_-a_+a_ & +a_+a_-a_-a_ \\ a_+a_-a_-a_ & +a_-a_+a_-a_ & -a_-a_-a_-a_ & -a_+a_+a_-a_ \end$

has rank one and is of unit Euclidean norm as a 16D vector if and only if is indeed a 4D rotation matrix. In this case there exist real numbers and such that

:$M= \begin ap & aq & ar & as \\ bp & bq & br & bs \\ cp & cq & cr & cs \\ dp & dq & dr & ds \end$

and
:$(ap)^2 + \cdots + (ds)^2 = \left(a^2 + b^2 + c^2 + d^2\right)\left(p^2 + q^2 + r^2 + s^2\right) = 1.$
There are exactly two sets of and such that and . They are each other's opposites.

The rotation matrix then equals
:$\beginA&= \begin ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\ bp+aq-dr+cs&-bq+ap+ds+cr&-br+as-dp-cq&-bs-ar-dq+cp\\ cp+dq+ar-bs&-cq+dp-as-br&-cr+ds+ap+bq&-cs-dr+aq-bp\\ dp-cq+br+as&-dq-cp-bs+ar&-dr-cs+bp-aq&-ds+cr+bq+ap\end \\&= \begin a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end \begin p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end .\end$

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, i.e. the central inversion.

Relation to quaternions

A point in 4-dimensional space with Cartesian coordinates may be represented by a quaternion .

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion . In matrix-vector language this is
:$\begin u'\\x'\\y'\\z' \end = \begin a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end \begin u\\x\\y\\z \end.$

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion , which is in matrix-vector form
:$\begin u'\\x'\\y'\\z' \end = \begin u\\x\\y\\z \end \begin p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end.$

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
:$u' + x'i + y'j + z'k = (a + bi + cj + dk)(u + xi + yj + zk)(p + qi + rj + sk),$

or, in symbolic form,
:$P' = Q_\mathrm P Q_\mathrm.\,$

According to the German mathematician Felix Klein this formula was already known to Cayley in 1854.

Quaternion multiplication is associative. Therefore,
:$P' = \left(Q_\mathrm P\right) Q_\mathrm = Q_\mathrm \left(P Q_\mathrm\right),\,$
which shows that left-isoclinic and right-isoclinic rotations commute.

The eigenvalues of 4D rotation matrices

The four eigenvalues of a 4D rotation matrix generally occur as two conjugate pairs of complex numbers 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) is identified with the subgroup of SO(4) consisting of the matrices
:$\begin 1 & \,\, 0 & \,\, 0 & \,\, 0 \\ 0 & a_ & a_ & a_ \\ 0 & a_ & a_ & a_ \\ 0 & a_ & a_ & a_ \end.$

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 for 3D rotations
:$\begin a_ & a_ & a_ \\ a_ & a_ & a_ \\ a_ & a_ & a_ \end = \begin a^2 + b^2 - c^2 - d^2 & 2(bc - ad)& 2(bd + ac) \\ 2(bc + ad) & a^2 - b^2 + c^2 -d^2 & 2(cd - ab) \\ 2(bd - ac) & 2(cd + ab) & a^2 - b^2 - c^2 + d^2 \end,$

which is the representation of the 3D rotation by its Euler–Rodrigues parameters: .

The corresponding quaternion formula , where , or, in expanded form:
:$x'i + y'j + z'k = (a + bi + cj + dk)(xi + yj + zk)(a - bi - cj - dk)$
is known as the Hamilton– Cayley formula.

Hopf coordinates

Rotations in 3D space are made mathematically much more tractable by the use of spherical coordinates. 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 referred to the fixed axis and invariant plane:
:$\begin x &= \sin\theta \cos \phi \\ y &= \sin\theta \sin \phi \\ z &= \cos\theta \end$

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:
:$\begin u &= \cos\xi_1 \sin\eta \\ z &= \sin\xi_1 \sin\eta \\ x &= \cos\xi_2 \cos\eta \\ y &= \sin\xi_2 \cos\eta \end$

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 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 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 and the Cayley formula. Let be a 4 × 4 skew-symmetric matrix. The skew-symmetric matrix can be uniquely decomposed as
:$A =\theta_1 A_1+\theta_2 A_2$
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
:$\left\.$
Then can be decomposed as
:$A=\theta_1 A_1+\theta_2 A_2$
where and are skew-symmetric matrices satisfying the properties
:$A_1 A_2=A_2 A_1=0, \qquad ^3=-A_1, \quad \text \quad ^3=-A_2.$

Moreover, the skew-symmetric matrices and are uniquely obtained as
:$A_1 = \frac$
and
:$A_2 = \frac.$

Then,
:$R = e^A = I + \sin\theta_1 A_1 + \left(1-\cos\theta_1\right) ^2 + \sin\theta_2 A_2 + \left(1-\cos\theta_2\right) ^2$
is a rotation matrix in , which is generated by Rodrigues' rotation formula, with the set of eigenvalues
:$\left\.$

Also,
:$R = (I+A)(I-A)^ = I+\fracA_1+\frac^2+\fracA_2+\frac^2$
is a rotation matrix in , which is generated by Cayley's rotation formula, such that the set of eigenvalues of is,
:$\left\.$

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
*Lorentz group
*Orthogonal group
*Orthogonal matrix
*Plane of rotation
*Poincaré group
*Quaternions and spatial rotation

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: 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 and Hilbert to four dimensions.
*J. H. Conway and D. A. Smith: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, 2003.
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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.
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*{{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