SO(4)

In mathematics, SO(4) is the four-dimensional rotation group; that is, the group of rotations about a fixed point in four-dimensional Euclidean space. The name comes from the fact that it is (isomorphic to) 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 $[0,π]$ except where mentioned or clearly implied by the context otherwise.

Geometry of 4D rotations

There are two kinds of 4D rotations: simple rotations and double rotations.

Simple rotations

A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) fixed. Every plane B that is completely orthogonal^[1] to A intersects A in a certain point P. Each such point P is the centre of the 2D rotation induced by R in B. All these 2D rotations have the same rotation angle $α$ .

Half-lines from O in the axis-plane A are not displaced; half-lines from O orthogonal to A are displaced through $α$ ; all other half-lines are displaced through an angle $< α$ .

Double rotations

A double rotation R about a rotation centre O leaves only O fixed. Any double rotation has at least one pair of completely orthogonal planes A and B through O that are invariant as a whole, i.e. rotated in themselves. In general the rotation angles $α$ in plane A and $β$ in plane B are different. In that case A and B are the only pair of invariant planes, and half-lines from O in A, B are displaced through $α$ , $β$ , and half-lines from O not in A or B are displaced through angles strictly between $α$ and $β$ .

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 O are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements. Beware: not all planes through O are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.

There are two kinds of isoclinic 4D rotations. To see this, consider an isoclinic rotation R, and take an ordered set OU, OX, OY, OZ of mutually perpendicular half-lines at O (denoted as OUXYZ) such that OU and OX span an invariant plane, and therefore OY and OZ also span an invariant plane. Now assume that only the rotation angle $α$ is specified. Then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle $α$ , depending on the rotation senses in OUX and OYZ.

We make the convention that the rotation senses from OU to OX and from OY to OZ are reckoned positive. Then we have the four rotations R1 = (+ $α$ , + $α$ ), R2 = (- $α$ , - $α$ ), R3 = (+ $α$ , - $α$ ) and R4 = (- $α$ , + $α$ ). R1 and R2 are each other's inverses; so are R3 and R4.

Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic. Left- (Right-) isoclinic rotations are represented by left- (right-) multiplication by unit quaternions; see the paragraph "Relation to quaternions" below.

The four rotations are pairwise different except if $α = 0$ or $α = π$ . $α = 0$ corresponds to the non-rotation; $α = π$ corresponds to the central inversion. These two elements of SO(4) are the only ones which are 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 R' with its own axes OU'X'Y'Z' is selected, then one can always choose the order of U', X', Y', Z' such that OUXYZ can be transformed into OU'X'Y'Z' by a rotation rather than by a rotation-reflection. Therefore, once one has selected a system OUXYZ 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 6-parameter Lie group.

Each plane through the rotation centre O 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 O is the pair of invariant planes of a commutative subgroup of SO(4) isomorphic to

SO(2)×SO(2).

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 S³_L of SO(4) which is isomorphic to the multiplicative group S³ of unit quaternions. All right-isoclinic rotations likewise form a subgroup S³_R of SO(4) isomorphic to S³. Both S³_L and S³_R are maximal subgroups of SO(4).

Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product S³_LxS³_R with normal subgroups S³_L and S³_R; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to S³.

Each 4D rotation R is in two ways the product of left- and right-isoclinic rotations R_L and R_R. R_L and R_R are together determined up to the central inversion, i.e. when both R_L and R_R are multiplied by the central inversion their product is R again.

This implies that S³_LxS³_R is the double cover of SO(4) and that S³_L and S³_R are normal subgroups of SO(4). The non-rotation I and the central inversion -I form a group C₂ of order 2, which is the centre of SO(4) and of both S³_L and S³_R. 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)xSO(3). The factor groups of C₂ in S³_L and S³_R are isomorphic to SO(3). The factor groups of S³_L and S³_R in SO(4) are isomorphic to SO(3).

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 -I and have the group C₂ = {I, -I} as their centre. From SO(6) onwards they are almost-simple in the sense that the factor groups of their centres are simple groups.

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 O the subgroups S³_L and S³_R are mutually conjugate and so are not 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 pair of isoclinic rotations through the same angle is conjugate. The sets of all isoclinic rotations are not even subgroups of SO(2N), let alone normal subgroups.

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 reals 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 $A= \begin{pmatrix} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \\ \end{pmatrix}$ be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called associate matrix

$M= \frac{1}{4} \begin{pmatrix} a_{00}+a_{11}+a_{22}+a_{33} & +a_{10}-a_{01}-a_{32}+a_{23} & +a_{20}+a_{31}-a_{02}-a_{13} & +a_{30}-a_{21}+a_{12}-a_{03} \\ a_{10}-a_{01}+a_{32}-a_{23} & -a_{00}-a_{11}+a_{22}+a_{33} & +a_{30}-a_{21}-a_{12}+a_{03} & -a_{20}-a_{31}-a_{02}-a_{13} \\ a_{20}-a_{31}-a_{02}+a_{13} & -a_{30}-a_{21}-a_{12}-a_{03} & -a_{00}+a_{11}-a_{22}+a_{33} & +a_{10}+a_{01}-a_{32}-a_{23} \\ a_{30}+a_{21}-a_{12}-a_{03} & +a_{20}-a_{31}+a_{02}-a_{13} & -a_{10}-a_{01}-a_{32}-a_{23} & -a_{00}+a_{11}+a_{22}-a_{33} \end{pmatrix}$

M has rank one and is of unit Euclidean norm as a 16D vector if and only if A is indeed a 4D rotation matrix. In this case there exist reals a, b, c, d; p, q, r, s such that

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

and $(a p) 2 + ... + (d s) 2 =$ $(a 2 + b 2 + c 2 + d 2)(p 2 + q 2 + r 2 + s 2) = 1$ . There are exactly two sets of a, b, c, d; p, q, r, s such that $a 2 + b 2 + c 2 + d 2 = 1$ and $p 2 + q 2 + r 2 + s 2 = 1$ . They are each other's opposites.

The rotation matrix then equals

$A= \begin{pmatrix} ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\ bp+aq-dr+cs&-bq+ap+ds+cr&-br+as-dp-cq&-bs-ar-dq+cp\\ cp+dq+ar-bs&-cq+dp-as-br&-cr+ds+ap+bq&-cs-dr+aq-bp\\ dp-cq+br+as&-dq-cp-bs+ar&-dr-cs+bp-aq&-ds+cr+bq+ap\end{pmatrix}$

$= \begin{pmatrix} a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end{pmatrix} \cdot \begin{pmatrix} p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end{pmatrix} .$

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 4D space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion u + xi + yj + zk.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Q_L = a + bi + cj + dk. In matrix-vector language this is

$\begin{pmatrix} u'\\x'\\y'\\z' \end{pmatrix} = \begin{pmatrix} a&-b&-c&-d\\ b&\;\,\, a&-d&\;\,\, c\\ c&\;\,\, d&\;\,\, a&-b\\ d&-c&\;\,\, b&\;\,\, a \end{pmatrix} \cdot \begin{pmatrix} u\\x\\y\\z \end{pmatrix}$

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion Q_R = p + qi + rj + sk, which is in matrix-vector form

$\begin{pmatrix} u'\\x'\\y'\\z' \end{pmatrix} = \begin{pmatrix} p&-q&-r&-s\\ q&\;\,\, p&\;\,\, s&-r\\ r&-s&\;\,\, p&\;\,\, q\\ s&\;\,\, r&-q&\;\,\, p \end{pmatrix} \cdot \begin{pmatrix} u\\x\\y\\z \end{pmatrix}.$

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

$P' = Q_L\cdot P\cdot Q_R.\,$

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

Quaternion multiplication is associative. Therefore

$P' = (Q_L\cdot P)\cdot Q_R = Q_L\cdot (P\cdot Q_R),\,$

which shows that left-isoclinic and right-isoclinic rotations commute.

In quaternion notation, a rotation in SO(4) is a single rotation if and only if Q_L and Q_R are conjugate elements of the group of unit quaternions. This is equivalent to the statement that Q_L and Q_R have the same real part, i.e. a = p.

The Euler–Rodrigues formula for 3D rotations

Our ordinary 3D space is conveniently treated as the subspace with coordinate system OXYZ of the 4D space with coordinate system OUXYZ. Its rotation group is identified with the subgroup of SO(4) consisting of the matrices

$\begin{pmatrix} 1 & \,\, 0 & \,\, 0 & \,\, 0 \\ 0 & a_{11} & a_{12} & a_{13} \\ 0 & a_{21} & a_{22} & a_{23} \\ 0 & a_{31} & a_{32} & a_{33} \end{pmatrix}.$

In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to p = a, q = −b, r = −c, s = −d, or in quaternion representation: Q_R = Q_L' = Q_L⁻¹. The 3D rotation matrix then becomes

$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{pmatrix} = \begin{pmatrix} a^2 + b^2 - c^2 - d^2 & 2(bc - ad)& 2(bd + ac) \\ 2(bc + ad) & a^2 - b^2 + c^2 -d^2 & 2(cd - ab) \\ 2(bd - ac) & 2(cd + ab) & a^2 - b^2 - c^2 + d^2 \end{pmatrix},$

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

The corresponding quaternion formula

P' = Q P Q - 1,

where Q = Q_L,

or, in expanded form:

x' i + y' j + z' k = (a + b i + c j + d k)(x i + y j + z k)(a - b i - c j - d k)

is known as the Hamilton–Cayley formula.

Notes

^ Two flat subspaces S1 and S2 of dimensions M and N of a Euclidean space S of at least M+N dimensions are called completely orthogonal if every line in S1 is orthogonal to every line in S2. If dim(S) = M+N then S1 and S2 intersect in a single point O. If dim(S) > M+N then S1 and S2 may or may not intersect. If dim(S) = M+N then a line in S1 and a line in S2 may or may not intersect; if they intersect then they intersect in O. Literature: Schoute 1902, Volume 1.

References

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.
Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations., arXiv General Mathematics 2005.
Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007.
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.