Rotations in 4-dimensional Euclidean space

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 (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 $[0, \pi]$ 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 $\alpha$.

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

Double rotations

Tesseract, in stereographic projection, in double rotation
A 4D clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as in helical path on that torus

For each rotation R of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes A and B each of which is carried onto itself by R and whose direct sum A⊕B is all of 4-space. Hence R restricted to either one of these is an ordinary rotation of a 2-plane. For almost all R (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles α in plane A and β in plane B — both assumed to be nonzero — are different. The unequal rotation angles α and β satisfying -π < α, β < π are almost* uniquely determined by R. Assuming that 4-space is oriented, then the orientations of the 2-planes A and B can be chosen consistent with this orientation in two ways. If the rotation angles are unequal (α ≠ β), R is sometimes termed a "double rotation".

In that case of a double rotation, A and B are the only pair of invariant planes, and half-lines from the origin in A, B are displaced through α and β respectively, and half-lines from the origin not in A or B are displaced through angles strictly between α and β.

*Assuming that 4-space is oriented, then an orientation for each of the 2-planes A and B can be chosen to be consistent with this orientation of 4-space in two equally valid ways. If the angles from one such choice of orientations of A and B are {α, β}, then the angles from the other choice are {-α, -β}. (In order to measure a rotation angle in a 2-plane, it is necessary to specify an orientation on that 2-plane. A rotation angle of -π is the same as one of +π. If the orientation of 4-space is reversed, the resulting angles would be either {α, -β} or {-α, β}. Hence the absolute values of the angles are well-defined completely independent of any choices.)

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 $\alpha$ is specified. Then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle $\alpha$, 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 = $(+\alpha, +\alpha)$, R2 = $(-\alpha, -\alpha)$, R3 = $(+\alpha, -\alpha)$ and R4 = $(-\alpha, +\alpha)$. 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 $\alpha = 0$ or $\alpha = \pi$. $\alpha = 0$ corresponds to the identity rotation; $\alpha = \pi$ 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 compact 6-dimensional 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 S3L of SO(4) which is isomorphic to the multiplicative group S3 of unit quaternions. All right-isoclinic rotations likewise form a subgroup S3R of SO(4) isomorphic to S3. Both S3L and S3R are maximal subgroups of SO(4).

Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product S3L × S3R with normal subgroups S3L and S3R; 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 RL and RR. RL and RR are together determined up to the central inversion, i.e. when both RL and RR are multiplied by the central inversion their product is R again.

This implies that S3L × S3R is the universal covering group of SO(4) — its unique double cover — and that S3L and S3R are normal subgroups of SO(4). The identity rotation I and the central inversion -I form a group C2 of order 2, which is the centre of SO(4) and of both S3L and S3R. 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 C2 in S3L and in S3R are each isomorphic to SO(3). The factor groups of S3L and of S3R in SO(4) are each isomorphic to SO(3).

The topology of SO(4) is the same as that of the Lie group SO(3) × Spin(3) = SO(3) × SU(2), namely the topology of P3 × S3. 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 SO(3) × Spin(3) = SO(3) × SU(2).

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 S3L and S3R 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 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 $A= \begin{pmatrix} a_{00} & a_{01} & a_{02} & a_{03} \\ a_{10} & a_{11} & a_{12} & a_{13} \\ a_{20} & a_{21} & a_{22} & a_{23} \\ a_{30} & a_{31} & a_{32} & a_{33} \\ \end{pmatrix}$ 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 $(ap)^2 + \cdots + (ds)^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 QL = 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 QR = 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 QL and QR are conjugate elements of the group of unit quaternions. This is equivalent to the statement that QL and QR have the same real part and opposite imaginary part, i.e. $a = p$, $b = -q$, $c = -r$, $d = -s$.

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 SO(3) 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: QR = QL' = QL−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: a, b, c, d.

The corresponding quaternion formula

$P' = QPQ^{-1}$,

where Q = QL, 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 HamiltonCayley formula.