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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 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 independently 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 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_L × S³_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_L × S³_R is the universal covering group of SO(4) — its unique double cover — and that S³_L and S³_R 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 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) × SO(3). The factor groups of C₂ in S³_L and in S³_R are each isomorphic to SO(3). The factor groups of S³_L and of S³_R 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 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.