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In [[mathematics]], the [[group (mathematics)|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 (mathematics)|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===
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]]<ref>Two flat subspaces ''S''<sub>1</sub> and ''S''<sub>2</sub> of dimensions ''M'' and ''N'' of a Euclidean space ''S'' of at least ''M''&nbsp;+&nbsp;''N'' dimensions are called ''completely orthogonal'' if every line in S1 is orthogonal to every line in S2. If dim(''S'')&nbsp;=&nbsp;''M''&nbsp;+&nbsp;''N'' then S1 and S2 intersect in a single point O. If dim(''S'')&nbsp;>&nbsp;''M''&nbsp;+&nbsp;''N'' then S1 and S2 may or may not intersect. If dim(''S'')&nbsp;=&nbsp;''M''&nbsp;+&nbsp;''N'' then a line in ''S''<sub>1</sub> and a line in ''S''</sub>2</sub> may or may not intersect; if they intersect then they intersect in O. Literature: Schoute 1902, Volume 1.</ref> 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 <math>\alpha</math>.

[[Half-line]]s from O in the axis-plane A are not displaced; half-lines from O orthogonal to A are displaced through <math>\alpha</math>; all other half-lines are displaced through an angle <math>< \alpha</math>.

===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 (mathematics)|invariant]] as a whole, i.e. rotated in themselves.
In general the rotation angles <math>\alpha</math> in plane A and <math>\beta</math> in plane B are different.
In that case A and B are the only pair of invariant planes, and [[half-line]]s from O in A, B are displaced through <math>\alpha</math>, <math>\beta</math>, and half-lines from O not in A or B are displaced through angles strictly between <math>\alpha</math> and <math>\beta</math>.

===Isoclinic rotations===
If the rotation angles of a double rotation are equal then there are infinitely many [[Invariant (mathematics)|invariant]] planes instead of just two, and all [[half-line]]s 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 <math>\alpha</math> is specified. Then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle <math>\alpha</math>, 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 = <math>(+\alpha, +\alpha)</math>, R2 = <math>(-\alpha, -\alpha)</math>, R3 = <math>(+\alpha, -\alpha)</math> and R4 = <math>(-\alpha, +\alpha)</math>. R1 and R2 are each other's [[inverse function|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 <math>\alpha = 0</math> or <math>\alpha = \pi</math>.
<math>\alpha = 0</math> corresponds to the identity rotation; <math>\alpha = \pi</math> corresponds to the [[inversion in a point|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 [[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)===
SO(4) is a [[noncommutative]] [[Compact_space|compact]] 6-[[Dimension#Manifolds|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 [[Conjugation of isometries in Euclidean space|conjugate]] in SO(4).

Each pair of completely [[orthogonal]] planes through O is the pair of [[Invariant (mathematics)|invariant]] planes of a commutative subgroup of SO(4) isomorphic to ''SO''(2) &times; ''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 S<sup>3</sup><sub>L</sub> of SO(4) which is isomorphic to the [[multiplication|multiplicative]] group S<sup>3</sup> of unit [[quaternion]]s. All right-isoclinic rotations likewise form a subgroup S<sup>3</sup><sub>R</sub> of SO(4) isomorphic to S<sup>3</sup>. Both S<sup>3</sup><sub>L</sub> and S<sup>3</sup><sub>R</sub> are maximal subgroups of SO(4).

Each left-isoclinic rotation [[commutative|commutes]] with each right-isoclinic rotation. This implies that there exists a [[direct product of groups|direct product]] S<sup>3</sup><sub>L</sub> &times; S<sup>3</sup><sub>R</sub> with [[normal subgroup]]s S<sup>3</sup><sub>L</sub> and S<sup>3</sup><sub>R</sub>; both of the corresponding [[factor group]]s are isomorphic to the other factor of the direct product, i.e. isomorphic to S<sup>3</sup>.

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

This implies that S<sup>3</sup><sub>L</sub> &times; S<sup>3</sup><sub>R</sub> is the [[universal covering group]] of SO(4) — its unique [[Double covering group|double cover]] — and that S<sup>3</sup><sub>L</sub> and S<sup>3</sup><sub>R</sub> are normal subgroups of SO(4). The identity rotation I and the central inversion -I form a group C<sub>2</sub> of order 2, which is the [[center of a group|centre]] of SO(4) and of both S<sup>3</sup><sub>L</sub> and S<sup>3</sup><sub>R</sub>. The centre of a group is a normal subgroup of that group. The factor group of C<sub>2</sub> in SO(4) is isomorphic to SO(3) &times; SO(3). The factor groups of C<sub>2</sub> in S<sup>3</sup><sub>L</sub> and in S<sup>3</sup><sub>R</sub> are each isomorphic to SO(3). The factor groups of S<sup>3</sup><sub>L</sub> and of S<sup>3</sup><sub>R</sub> in SO(4) are each 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 group]]s.

The even-dimensional rotation groups do contain the central inversion −I and have the group C<sub>2</sub> = {I, −I} as their [[center of a group|centre]]. From SO(6) onwards they are almost-simple in the sense that the [[factor group]]s of their centres are simple groups.

SO(4) is different: there is no [[Conjugation of isometries in Euclidean space|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 S<sup>3</sup><sub>L</sub> and S<sup>3</sup><sub>R</sub> 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 (vector space)|orientation]]-preserving [[Isometry|isometric]] [[linear]] mappings of a 4D [[vector space]] with [[inner product]] over the [[real number]]s onto itself.

With respect to an [[orthonormal]] [[basis (linear algebra)|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
<math>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}
</math>
be its matrix with respect to an arbitrary orthonormal basis.

Calculate from this the so-called ''associate matrix''

<math>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}
</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

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

and <math>(ap)^2 + \cdots + (ds)^2 = </math><math>(a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) = 1</math>.
There are exactly two sets of a, b, c, d; p, q, r, s such that <math>a^2 + b^2 + c^2 + d^2 = 1</math> and <math>p^2 + q^2 + r^2 + s^2 = 1</math>. They are each other's opposites.

The rotation matrix then equals

<math>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}
</math>

<math>
=
\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}
.
</math>

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<sub>L</sub> = a + bi + cj + dk. In matrix-vector language this is

<math>
\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}
</math>

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion Q<sub>R</sub> = p + qi + rj + sk, which is in matrix-vector form

<math>
\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}.
</math>

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

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

or in symbolic form

:<math>P' = Q_L\cdot P\cdot Q_R.\, </math>

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

Quaternion multiplication is [[associative]].  Therefore

:<math>P' = (Q_L\cdot P)\cdot Q_R = Q_L\cdot (P\cdot Q_R),\,</math>

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<sub>L</sub> and  Q<sub>R</sub> are conjugate elements of the group of unit quaternions. This is equivalent to the statement that Q<sub>L</sub> and  Q<sub>R</sub> have the same real part, i.e. <math>a = p</math>.

===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

: <math>
\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}.
</math>

In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to <math>p = a, q = -b, r = -c, s = -d</math>, or in quaternion representation: ''Q''<sub>''R''</sub> = ''Q''<sub>''L''</sub><nowiki>'</nowiki> = ''Q''<sub>''L''</sub><sup>&minus;1</sup>.
The 3D rotation matrix then becomes

: <math>
\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},
</math>

which is the representation of the 3D rotation by its [[Euler–Rodrigues parameters]]: a, b, c, d.

The corresponding quaternion formula

<math>P' = QPQ^{-1}</math>, (<math>P'</math> is / means ''P - prime'')

where ''Q'' = ''Q''<sub>L,</sub> or, in expanded form:

:<math>x'i + y'j + z'k = (a + bi + cj + dk)(xi + yj + zk)(a - bi - cj - dk)</math>

is known as the [[William Rowan Hamilton|Hamilton]]–[[Arthur Cayley|Cayley]] formula.

==See also==
*[[orthogonal matrix]]
*[[orthogonal group]]
*[[Lorentz group]]
*[[Poincaré group]]
*[[Laplace–Runge–Lenz vector]]
*[[Plane of rotation]]
*[[Quaternions and spatial rotation]]

==Notes==
{{Reflist}}

==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.
*[http://www.brown.edu/Administration/News_Bureau/Databases/Encyclopedia/search.php?serial=M0090 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 [[David Hilbert|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, [http://arxiv.org/abs/math/0501249  A matrix-based proof of the quaternion representation theorem for four-dimensional rotations.], ''arXiv General Mathematics'' 2005.
*Johan E. Mebius, [http://arxiv.org/abs/math/0701759  Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations.], ''arXiv General Mathematics'' 2007.
*[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.
{{Use dmy dates|date=September 2010}}

[[Category:Rotation]]
[[Category:Four-dimensional geometry]]
[[Category:Quaternions]]