View source for Rotations in 4-dimensional Euclidean space

{{Use dmy dates|date=September 2010}}
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 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 {{closed-closed|0, π}} except where mentioned or clearly implied by the context otherwise.

A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation.

==Geometry of 4D rotations==
Four-dimensional rotations are of two types: simple rotations and double rotations.

===Simple rotations===
A simple rotation {{mvar|R}} about a rotation centre {{mvar|O}} leaves an entire plane {{mvar|A}} through {{mvar|O}} (axis-plane) fixed. Every plane {{mvar|B}} that is completely [[orthogonality|orthogonal]]{{efn|group=nb|Two flat subspaces {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}} of dimensions {{mvar|M}} and {{mvar|N}} of a Euclidean space {{mvar|S}} of at least {{math|''M'' + ''N''}} dimensions are called ''completely orthogonal'' if every line in {{math|''S''<sub>1</sub>}} is orthogonal to every line in {{math|''S''<sub>2</sub>}}. If {{math|dim(''S'') {{=}} ''M'' + ''N''}} then {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}} intersect in a single point {{mvar|O}}. If {{math|dim(''S'') > ''M'' + ''N''}} then {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}} may or may not intersect. If {{math|dim(''S'') {{=}} ''M'' + ''N''}} then a line in {{math|''S''<sub>1</sub>}} and a line in {{math|''S''<sub>2</sub>}} may or may not intersect; if they intersect then they intersect in O.<ref>Schoute 1902, Volume 1.</ref>}} to {{mvar|A}} intersects {{mvar|A}} in a certain point {{mvar|P}}. Each such point {{mvar|P}} is the centre of the 2D rotation induced by {{mvar|R}} in {{mvar|B}}. All these 2D rotations have the same rotation angle {{mvar|α}}.

[[Half-line]]s from {{mvar|O}} in the axis-plane {{mvar|A}} are not displaced; half-lines from {{mvar|O}} orthogonal to {{mvar|A}} are displaced through {{mvar|α}}; all other half-lines are displaced through an angle less than {{mvar|α}}.

===Double rotations===
[[File:Tesseract.gif|thumb|[[Tesseract]], in [[stereographic projection]], in '''double rotation''']]
[[File:Torus vectors oblique.jpg|thumb|left|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 a rotation whose two rotation angles form a rational number, the paths will eventually reconnect, while for an irrational ratio they will not. An isoclinic rotation will form a [[Villarceau circle]] on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis.]]
For each rotation {{mvar|R}} of 4-space (fixing the origin), there is at least one pair of [[orthogonality|orthogonal]] 2-planes {{mvar|A}} and {{mvar|B}} each of which is invariant and whose direct sum {{math|''A'' ⊕ ''B''}} is all of 4-space. Hence {{mvar|R}} operating on either of these planes produces an ordinary rotation of that plane. For almost all {{mvar|R}} (all of the 6-dimensional set of rotations except for a 3-dimensional subset), the rotation angles {{mvar|α}} in plane {{mvar|A}} and {{mvar|β}} in plane {{mvar|B}} — both assumed to be nonzero — are different. The unequal rotation angles {{mvar|α}} and {{mvar|β}} satisfying {{math|−π < ''α''}}, {{math|''β'' < π}} are almost{{efn|group=nb|Assuming that 4-space is oriented, then an orientation for each of the 2-planes {{mvar|A}} and {{mvar|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 {{mvar|A}} and {{mvar|B}} are {{math|{''α'', ''β''<nowiki>}</nowiki>}}, then the angles from the other choice are {{math|{−''α'', −''β''<nowiki>}</nowiki>}}. (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 −{{pi}} is the same as one of +{{pi}}. If the orientation of 4-space is reversed, the resulting angles would be either {{math|{''α'', −''β''<nowiki>}</nowiki>}} or {{math|{−''α'', ''β''<nowiki>}</nowiki>}}. Hence the '''absolute values''' of the angles are well-defined completely independently of any choices.)}} uniquely determined by {{mvar|R}}. Assuming that 4-space is oriented, then the orientations of the 2-planes {{mvar|A}} and {{mvar|B}} can be chosen consistent with this orientation in two ways. If the rotation angles are unequal ({{math|''α'' ≠ ''β''}}), {{mvar|R}} is sometimes termed a "double rotation".

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

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

Assuming that a fixed orientation has been chosen for 4-dimensional space, isoclinic 4D rotations may be put into two categories. To see this, consider an isoclinic rotation {{mvar|R}}, and take an orientation-consistent ordered set {{math|''OU'', ''OX'', ''OY'', ''OZ''}} of mutually perpendicular half-lines at {{mvar|O}} (denoted as {{mvar|OUXYZ}}) such that {{mvar|OU}} and {{mvar|OX}} span an invariant plane, and therefore {{mvar|OY}} and {{mvar|OZ}} also span an invariant plane. Now assume that only the rotation angle {{mvar|α}} is specified. Then there are in general four isoclinic rotations in planes {{mvar|OUX}} and {{mvar|OYZ}} with rotation angle {{mvar|α}}, depending on the rotation senses in {{mvar|OUX}} and {{mvar|OYZ}}.

We make the convention that the rotation senses from {{mvar|OU}} to {{mvar|OX}} and from {{mvar|OY}} to {{mvar|OZ}} are reckoned positive. Then we have the four rotations {{math|''R''<sub>1</sub> {{=}} (+''α'', +''α'')}}, {{math|''R''<sub>2</sub> {{=}} (−''α'', −''α'')}}, {{math|''R''<sub>3</sub> {{=}} (+''α'', −''α'')}} and {{math|''R''<sub>4</sub> {{=}} (−''α'', +''α'')}}. {{math|''R''<sub>1</sub>}} and {{math|''R''<sub>2</sub>}} are each other's [[inverse function|inverses]]; so are {{math|''R''<sub>3</sub>}} and {{math|''R''<sub>4</sub>}}. As long as {{mvar|α}} lies between 0 and {{pi}}, these four rotations will be distinct.

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

The four rotations are pairwise different except if {{math|''α'' {{=}} 0}} or {{math|''α'' {{=}} π}}. The angle {{math|''α'' {{=}} 0}} corresponds to the identity rotation; {{math|''α'' {{=}} π}} corresponds to the [[inversion in a point|central inversion]], given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously 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 {{mvar|R′}} with its own axes {{mvar|OU′}}, {{mvar|OX′}}, {{mvar|OY′}}, {{mvar|OZ′}} is selected, then one can always choose the [[even permutation|order]] of {{mvar|U′}}, {{mvar|X′}}, {{mvar|Y′}}, {{mvar|Z′}} such that {{mvar|OUXYZ}} can be transformed into {{mvar|OU′X′Y′Z′}} by a rotation rather than by a rotation-reflection (that is, so that the ordered basis {{mvar|OU′}}, {{mvar|OX′}}, {{mvar|OY′}}, {{mvar|OZ′}} is also consistent with the same fixed choice of orientation as {{mvar|OU}}, {{mvar|OX}}, {{mvar|OY}}, {{mvar|OZ}}). Therefore, once one has selected an orientation (that is, a system {{mvar|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 {{mvar|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 [[orthogonality|orthogonal]] planes through {{mvar|O}} is the pair of [[invariant (mathematics)|invariant]] planes of a commutative subgroup of SO(4) isomorphic to {{nowrap|SO(2) × SO(2)}}.

These groups are [[maximal torus|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 {{math|''S''}}<sup>3</sup><sub>L</sub> of SO(4), which is isomorphic to the [[multiplicative group]] {{math|''S''}}<sup>3</sup> of unit [[quaternion]]s. All right-isoclinic rotations likewise form a subgroup {{math|''S''}}<sup>3</sup><sub>R</sub> of SO(4) isomorphic to {{math|''S''}}<sup>3</sup>. Both {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''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]] {{nowrap|{{math|''S''}}<sup>3</sup><sub>L</sub> × {{math|''S''}}<sup>3</sup><sub>R</sub>}} with [[normal subgroup]]s {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''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 {{math|''S''}}<sup>3</sup>. (This is not SO(4) or a subgroup of it, because {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''S''}}<sup>3</sup><sub>R</sub> are not disjoint: the identity {{mvar|I}} and the central inversion {{math|−''I''}} each belong to both {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''S''}}<sup>3</sup><sub>R</sub>.)

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

This implies that {{math|''S''}}<sup>3</sup><sub>L</sub> × {{math|''S''}}<sup>3</sup><sub>R</sub> is the [[universal covering group]] of SO(4) — its unique [[double covering group|double cover]] — and that {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''S''}}<sup>3</sup><sub>R</sub> are normal subgroups of SO(4). The identity rotation {{mvar|I}} and the central inversion {{math|−''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 {{math|''S''}}<sup>3</sup><sub>L</sub> and {{math|''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)&nbsp;×&nbsp;SO(3). The factor groups of {{math|''S''}}<sup>3</sup><sub>L</sub> by C<sub>2</sub> and of {{math|''S''}}<sup>3</sup><sub>R</sub> by C<sub>2</sub> are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by {{math|''S''}}<sup>3</sup><sub>L</sub> and of SO(4) by {{math|''S''}}<sup>3</sup><sub>R</sub> are each isomorphic to SO(3).

The '''topology''' of SO(4) is the same as that of the Lie group SO(3)&nbsp;×&nbsp;Spin(3) = SO(3)&nbsp;×&nbsp;SU(2), namely the topology of {{math|''P''<sup>3</sup> × ''S''<sup>3</sup>}}{{Definition needed|date=October 2019}}. 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 {{nowrap|1=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 group]]s.

The even-dimensional rotation groups do contain the central inversion {{math|−''I''}} and have the group {{nowrap|1=C<sub>2</sub> = <nowiki>{</nowiki>{{math|''I''}}, {{math|−''I''}}<nowiki>}</nowiki>}} 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 {{mvar|O}} the subgroups {{math|''S''<sup>3</sup><sub>L</sub>}} and {{math|''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(2{{math|''N''}}), 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 matrix|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''{{Citation needed|reason=Who calls it so?|date=August 2019}}

:<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>

{{mvar|M}} has [[rank (linear algebra)|rank]] one and is of unit [[Euclidean norm]] as a 16D vector if and only if {{mvar|A}} is indeed a 4D rotation matrix{{Citation needed|reason=Where can we see a discussion of this?|date=August 2019}}. In this case there exist real numbers {{math|''a'', ''b'', ''c'', ''d''}} and {{math|''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 = \left(a^2 + b^2 + c^2 + d^2\right)\left(p^2 + q^2 + r^2 + s^2\right) = 1.</math>{{Citation needed|reason=Please provide argument or reference.|date=August 2019}}
There are exactly two sets of {{math|''a'', ''b'', ''c'', ''d''}} and {{math|''p'', ''q'', ''r'', ''s''}} such that {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> + ''c''<sup>2</sup> + ''d''<sup>2</sup> {{=}} 1}} and {{math|''p''<sup>2</sup> + ''q''<sup>2</sup> + ''r''<sup>2</sup> + ''s''<sup>2</sup> {{=}} 1}}. They are each other's opposites.

The rotation matrix then equals
:<math>\begin{align}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}
\begin{pmatrix}
p&-q&-r&-s\\
q&\;\,\, p&\;\,\, s&-r\\
r&-s&\;\,\, p&\;\,\, q\\
s&\;\,\, r&-q&\;\,\, p
\end{pmatrix}
.\end{align}
</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 4-dimensional space with [[Cartesian coordinates]] {{math|(''u'', ''x'', ''y'', ''z'')}} may be represented by a [[quaternion]] {{math|1=''P'' = ''u'' + ''xi'' + ''yj'' + ''zk''}}.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion {{math|1=''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}
\begin{pmatrix}
u\\x\\y\\z
\end{pmatrix}
</math>

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion {{math|1=''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}
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}.
</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_\mathrm{L} P Q_\mathrm{R}.\, </math>

According to the German mathematician [[Felix Klein]] this formula was already known to Cayley in 1854{{Citation needed|reason=Reference needed|date=August 2019}}.

Quaternion multiplication is [[associative]]. Therefore,
:<math>P' = \left(Q_\mathrm{L} P\right) Q_\mathrm{R} = Q_\mathrm{L} \left(P Q_\mathrm{R}\right),\,</math>
which shows that left-isoclinic and right-isoclinic rotations commute.

===The eigenvalues of 4D rotation matrices===
The four [[eigenvalue]]s of a 4D rotation matrix generally occur as two conjugate pairs of [[complex numbers]] of unit magnitude. If an eigenvalue is real, it must be ±1, since a rotation leaves the magnitude of a vector unchanged. The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions {{math|''Q''<sub>L</sub>}} and {{math|''Q''<sub>R</sub>}} are equal in magnitude and have the same sign.{{efn|group=nb|Example of opposite signs: the central inversion; in the quaternion representation the real parts are +1 and −1, and the central inversion cannot be accomplished by a single simple rotation.}} If they are both zero, all eigenvalues of the rotation are unity, and the rotation is the null rotation. If the real parts of {{math|''Q''<sub>L</sub>}} and {{math|''Q''<sub>R</sub>}} are not equal then all eigenvalues are complex, and the rotation is a double rotation.

===The Euler–Rodrigues formula for 3D rotations===
Our ordinary 3D space is conveniently treated as the subspace with coordinate system 0XYZ of the 4D space with coordinate system UXYZ. 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''}}, {{math|''q'' {{=}} −''b''}}, {{math|''r'' {{=}} −''c''}}, {{math|''s'' {{=}} −''d''}}, or in quaternion representation: {{math|''Q''<sub>''R''</sub> {{=}} ''Q''<sub>''L''</sub>′ {{=}} ''Q''<sub>''L''</sub><sup>−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]]: {{math|''a'', ''b'', ''c'', ''d''}}.

The corresponding quaternion formula {{math|''P′'' {{=}} ''QPQ''<sup>−1</sup>}}, where {{math|''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.

===Hopf coordinates===
 
Rotations in 3D space are made mathematically much more tractable by the use of [[hyperspherical coordinates|spherical coordinates]]. Any rotation in 3D can be characterized by a fixed axis of rotation and an invariant plane perpendicular to that axis. Without loss of generality, we can take the {{mvar|xy}}-plane as the invariant plane and the {{mvar|z}}-axis as the fixed axis. Since radial distances are not affected by rotation, we can characterize a rotation by its effect on the unit sphere (2-sphere) by [[spherical coordinates]] referred to the fixed axis and invariant plane:
:<math>\begin{align}
x &= \sin\theta \cos \phi \\
y &= \sin\theta \sin \phi \\
z &= \cos\theta
\end{align}</math>

Because {{math|''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> {{=}} 1}}, the points lie on the 2-sphere. A point at {{math|{''θ''<sub>0</sub>, ''φ''<sub>0</sub><nowiki>}</nowiki>}} rotated by an angle {{mvar|φ}} about the {{mvar|z}}-axis is specified simply by {{math|{''θ''<sub>0</sub>, ''φ''<sub>0</sub> + ''φ''<nowiki>}</nowiki>}}. While [[hyperspherical coordinates]] are also useful in dealing with 4D rotations, an even more useful coordinate system for 4D is provided by [[3-sphere#Hopf coordinates|Hopf coordinates]] {{math|{''ξ''<sub>1</sub>, ''η'', ''ξ''<sub>2</sub><nowiki>}</nowiki>}},<ref name="Karcher">{{Citation
|url= http://virtualmathmuseum.org/Surface/bianchi-pinkall_tori/bianchi-pinkall_tori.html
|title= Bianchi–Pinkall Flat Tori in S<sub>3</sub>
|work= 3DXM Documentation
|last= Karcher |first= Hermann
|publisher= 3DXM Consortium
|accessdate= April 5, 2015
|ref= none
}}
</ref> which are a set of three angular coordinates specifying a position on the 3-sphere. For example:
:<math>\begin{align}
u &= \cos\xi_1 \sin\eta \\
z &= \sin\xi_1 \sin\eta \\
x &= \cos\xi_2 \cos\eta \\
y &= \sin\xi_2 \cos\eta
\end{align}</math>

Because {{math|''u''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> {{=}} 1}}, the points lie on the 3-sphere.

In 4D space, every rotation about the origin has two invariant planes which are completely orthogonal to each other and intersect at the origin, and are rotated by two independent angles {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}}. Without loss of generality, we can choose, respectively, the {{mvar|uz}}- and {{mvar|xy}}-planes as these invariant planes. A rotation in 4D of a point {{math|{''ξ''<sub>10</sub>, ''η''<sub>0</sub>, ''ξ''<sub>20</sub><nowiki>}</nowiki>}} through angles {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}} is then simply expressed in Hopf coordinates as {{math|{''ξ''<sub>10</sub> + ''ξ''<sub>1</sub>, ''η''<sub>0</sub>, ''ξ''<sub>20</sub> + ''ξ''<sub>2</sub><nowiki>}</nowiki>}}.

==Visualization of 4D rotations==
[[File:4DRotationTrajectories.jpg|thumb|upright=1.5|Trajectories of a point on the Clifford Torus:<br>Fig.1: simple rotations (black) and left and right isoclinic rotations (red and blue)<br>
Fig.2: a general rotation with angular displacements in a ratio of 1:5<br>
Fig.3: a general rotation with angular displacements in a ratio of 5:1<br>
All images are [[stereographic projection]]s.]]

Every rotation in 3D space has an invariant axis-line which is unchanged by the rotation. The rotation is completely specified by specifying the axis of rotation and the angle of rotation about that axis. Without loss of generality, this axis may be chosen as the {{mvar|z}}-axis of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 3D space, the [[spherical coordinates]] {{math|{''θ'', ''φ''<nowiki>}</nowiki>}} may be seen as a parametric expression of the 2-sphere. For fixed {{mvar|θ}} they describe circles on the 2-sphere which are perpendicular to the {{mvar|z}}-axis and these circles may be viewed as trajectories of a point on the sphere. A point {{math|{''θ''<sub>0</sub>, ''φ''<sub>0</sub><nowiki>}</nowiki>}} on the sphere, under a rotation about the {{mvar|z}}-axis, will follow a trajectory {{math|{''θ''<sub>0</sub>, ''φ''<sub>0</sub> + ''φ''<nowiki>}</nowiki>}} as the angle {{mvar|φ}} varies. The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: {{math|''φ'' {{=}} ''ωt''}}, with {{mvar|ω}} being an "angular velocity".

Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a point). The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the {{mvar|uz}}- and {{mvar|xy}}-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation.

In 4D space, the Hopf angles {{math|{''ξ''<sub>1</sub>, ''η'', ''ξ''<sub>2</sub><nowiki>}</nowiki>}} parameterize the 3-sphere. For fixed {{mvar|η}} they describe a torus parameterized by {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}}, with {{math|''η'' {{=}} {{sfrac|π|4}}}} being the special case of the [[Clifford torus]] in the {{mvar|xy}}- and {{mvar|uz}}-planes. These tori are not the usual tori found in 3D-space. While they are still 2D surfaces, they are embedded in the 3-sphere. The 3-sphere can be [[stereographic projection|stereographically]] projected onto the whole Euclidean 3D-space, and these tori are then seen as the usual tori of revolution. It can be seen that a point specified by {{math|{''ξ''<sub>10</sub>, ''η''<sub>0</sub>, ''ξ''<sub>20</sub><nowiki>}</nowiki>}} undergoing a rotation with the {{mvar|uz}}- and {{mvar|xy}}-planes invariant will remain on the torus specified by {{math|''η''<sub>0</sub>}}.<ref name="Pinkall">{{cite journal |last=Pinkall |first=U. |date=1985 |title=Hopf tori in S<sub>3</sub> |url=https://www.maths.ed.ac.uk/~aar/papers/pinkall.pdf |journal=Invent. Math. |publisher= |volume=81 |pages=379–386 |access-date=7 April 2015 |doi=10.1007/bf01389060}}</ref> The trajectory of a point can be written as a function of time as {{math|{''ξ''<sub>10</sub> + ''ω''<sub>1</sub>''t'', ''η''<sub>0</sub>, ''ξ''<sub>20</sub> + ''ω''<sub>2</sub>''t''<nowiki>}</nowiki>}} and stereographically projected onto its associated torus, as in the figures below.<ref name="Banchoff">{{cite book |last=Banchoff |first=Thomas F. |date=1990 |title=Beyond the Third Dimension |url=https://archive.org/details/beyondthirddimen00thom |location= |publisher=W H Freeman & Co; |isbn=978-0716750253 |access-date=2015-04-08 |url-access=registration }}</ref> In these figures, the initial point is taken to be {{math|{0, {{sfrac|π|4}}, 0<nowiki>}</nowiki>}}, i.e. on the Clifford torus. In Fig. 1, two simple rotation trajectories are shown in black, while a left and a right isoclinic trajectory is shown in red and blue respectively. In Fig. 2, a general rotation in which {{math|''ω''<sub>1</sub> {{=}} 1}} and {{math|''ω''<sub>2</sub> {{=}} 5}} is shown, while in Fig. 3, a general rotation in which {{math|''ω''<sub>1</sub> {{=}} 5}} and {{math|''ω''<sub>2</sub> {{=}} 1}} is shown.

==Generating 4D rotation matrices==
Four-dimensional rotations can be derived from [[Rodrigues' rotation formula]] and the Cayley formula. Let {{mvar|A}} be a 4&nbsp;×&nbsp;4 [[skew-symmetric matrix]]. The skew-symmetric matrix {{mvar|A}} can be uniquely decomposed as
:<math>A =\theta_1 A_1+\theta_2 A_2</math>
into two skew-symmetric matrices {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} satisfying the properties {{math|''A''<sub>1</sub>''A''<sub>2</sub> {{=}} 0}}, {{math|''A''<sub>1</sub><sup>3</sup> {{=}} −''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub><sup>3</sup> {{=}} −''A''<sub>2</sub>}}, where {{math|∓''θ''<sub>1</sub>''i''}} and {{math|∓''θ''<sub>2</sub>''i''}} are the eigenvalues of {{mvar|A}}. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} by Rodrigues' rotation formula and the Cayley formula.<ref>{{cite journal |last1=Erdoğdu |first1=M. |last2=Özdemir |first2=M. |date=2015 |title=Generating Four Dimensional Rotation Matrices |url=https://www.researchgate.net/publication/283007638_Generating_Four_Dimensional_Rotation_Matrices}}</ref>

Let {{mvar|A}} be a 4&nbsp;×&nbsp;4 nonzero skew-symmetric matrix with the set of eigenvalues
:<math>\left\{\theta_1 i,-\theta_1 i,\theta_2 i,-\theta_2 i : {\theta_1}^2 + {\theta_2}^2 > 0\right\}.</math>
Then {{mvar|A}} can be decomposed as
:<math>A=\theta_1 A_1+\theta_2 A_2</math>
where {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} are skew-symmetric matrices satisfying the properties
:<math>A_1 A_2=A_2 A_1=0, \qquad {A_1}^3=-A_1, \quad \text{and} \quad {A_2}^3=-A_2.</math>

Moreover, the skew-symmetric matrices {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} are uniquely obtained as
:<math>A_1 = \frac{{\theta_2}^2 A + A^3}{\theta_1 \left({\theta_2}^2 - {\theta_1}^2\right)}</math>
and
:<math>A_2 = \frac{{\theta_1}^2 A + A^3}{\theta_2 \left({\theta_1}^2 - {\theta_2}^2\right)}.</math>

Then,
:<math>R = e^A = I + \sin\theta_1 A_1 + \left(1-\cos\theta_1\right) {A_1}^2 + \sin\theta_2 A_2 + \left(1-\cos\theta_2\right) {A_2}^2</math>
is a rotation matrix in {{math|'''E'''<sup>4</sup>}}, which is generated by Rodrigues' rotation formula, with the set of eigenvalues
:<math>\left\{e^{\theta_1 i}, e^{-\theta_1 i}, e^{\theta_2 i}, e^{-\theta_2 i}\right\}.</math>

Also,
:<math>R = (I+A)(I-A)^{-1} = I+\frac{2\theta_1}{1+{\theta_1}^2}A_1+\frac{2{\theta_1}^2}{1+{\theta_1}^2}{A_1}^2+\frac{2\theta_2}{1+{\theta_2}^2}A_2+\frac{2{\theta_2}^2}{1+{\theta_2}^2}{A_2}^2</math>
is a rotation matrix in {{math|'''E'''<sup>4</sup>}}, which is generated by Cayley's rotation formula, such that the set of eigenvalues of {{mvar|R}} is,
:<math>\left\{\frac{\left(1+\theta_1 i\right)^2}{1+{\theta_1}^2},\frac{\left(1-\theta_1 i\right)^2}{1+{\theta_1}^2},\frac{\left(1+\theta_2 i\right)^2}{1+{\theta_2}^2},\frac{\left(1-\theta_2 i\right)^2}{1+{\theta_2}^2}\right\}.</math>

The generating rotation matrix can be classified with respect to the values {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} as follows:
# If {{math|''θ''<sub>1</sub> {{=}} 0}} and {{math|''θ''<sub>2</sub> ≠ 0}} or vice versa, then the formulae generate simple rotations;
# If {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} are nonzero and {{math|''θ''<sub>1</sub> ≠ ''θ''<sub>2</sub>}}, then the formulae generate double rotations;
# If {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} are nonzero and {{math|''θ''<sub>1</sub> {{=}} ''θ''<sub>2</sub>}}, then the formulae generate isoclinic rotations.

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

==Notes==
{{Notelist|group=nb}}

==References==
{{Reflist}}

==Bibliography==
*L. van Elfrinkhof: [https://archive.org/stream/handelingenvanh02unkngoog/#page/n289/mode/2up/search/237 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.
*[[Arthur Stafford Hathaway]] (1902) [https://www.jstor.org/stable/1986315 Quaternion Space], [[Transactions of the American Mathematical Society]] 3(1):46–59.
*Johan E. Mebius, [https://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, [https://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.
*{{cite journal |last1= Stringham |first1= Irving |authorlink= Irving Stringham |year= 1901 |title= On the geometry of planes in a parabolic space of four dimensions |journal= Transactions of the American Mathematical Society |volume= 2 |issue= |pages= 183–214 |jstor=1986218 |doi= 10.1090/s0002-9947-1901-1500564-2}}
*Melek Erdoğdu, Mustafa Özdemir, Generating Four Dimensional Rotation Matrices, https://www.researchgate.net/publication/283007638_Generating_Four_Dimensional_Rotation_Matrices, 2015.
*Daniele Mortari, "On the Rigid Rotation Concept in n-Dimensional Spaces", ''Journal of the Astronautical Sciences 49.3 (July 2001),'' https://pdfs.semanticscholar.org/f7d8/63ceb75277133592ef9e92457b6705b1264f.pdf

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