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In [[mathematics]], '''SO(4)''' is the
In this article ''rotation'' means ''rotational displacement''. For the sake of uniqueness rotation angles are assumed to be in the segment <math>[0, \pi]</math> 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 ===
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[[Half-line]]s from O in the axis-plane A are not displaced; half-lines from 0 orthogonal to A are displaced through <math>\alpha</math>; all other half-lines are displaced through an angle <math>< \alpha</math>.
(*) 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.
=== Double rotations ===
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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 [[Even_permutation |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) ===
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This implies that S3LxS3R is the [[double cover]] of SO(4) and that S3L and S3R 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 [[center of a group | 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)xSO(3). The factor groups of C2 in S3L and S3R are isomorphic to SO(3).
The factor groups of S3L and S3R in SO(4) are isomorphic to SO(3).
=== Special property of SO(4) among rotation groups in general ===
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SO(4) is different: there is no [[conjugation]] by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. [[Reflection_(mathematics)| Reflection]]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 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 ==
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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 ===
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Let
▲A=<math>
\begin{pmatrix}
a_{00} & a_{01} & a_{02} & a_{03} \\
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\end{pmatrix}
</math>
be its matrix with respect to an arbitrary orthonormal basis.
Calculate from this the so-called ''associate matrix''
▲M=<math>
\frac{1}{4}
\begin{pmatrix}
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\end{pmatrix}
</math>
M has [[Rank_(linear_algebra) | rank]] one and is of unit [[Norm_(mathematics) | 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
\begin{pmatrix}
ap & aq & ar & as \\
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The rotation matrix then equals
\begin{pmatrix}
ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\
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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 [[
=== 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.
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Quaternion multiplication is [[associative]]. Therefore P' = (QL.P).QR = QL.(P.QR), which shows that left-isoclinic and right-isoclinic rotations commute.
=== 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
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==See also==
*[[orthogonal matrix]]
*[[orthogonal group]]
*[[rotation group]]
*[[Lorentz group]]
== References ==
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[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schoute.html 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.
[[Category:Lie groups]]
[[Category:Quaternions]]
[[Category:4-dimensional geometry]]
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