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{{short description|Special orthogonal group}}

In [[mathematics]], '''SO(4)''' is the [[four-dimensional]] rotation group; that is, the [[group (mathematics)|group]] of [[rotations]] about a fixed point in four-dimensional [[Euclidean space]]. The name comes from the fact that it is ([[isomorphic]] to) the [[special orthogonal group]] of degree 4.

{{Use dmy dates|date=January 2020}}

In [[mathematics]], the [[group (mathematics)|group]] of '''rotations about a fixed point in [[four-dimensional space|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 <math>[{{closed-closed|0, \pi]</math>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 ==

There are two kinds of 4D rotations: simple rotations and double rotations.

==Geometry of 4D rotations==

=== Simple rotations ===

Four-dimensional rotations are of two types: simple rotations and double rotations.

A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) pointwise [[invariant]]. 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 <math>\alpha</math>.

===Simple rotations===

[[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>.

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 orthogonal]] to {{mvar|A}} intersects {{mvar|A}} in a certain point {{mvar|P}}. For 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}}.

[[Ray (geometry)|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}}.

(*) 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 ===

[[File:Tesseract.gif|thumb|[[Tesseract]], in [[stereographic projection]], in '''double rotation''']]

A double rotation R about a rotation centre O leaves only O [[invariant]]. Any double rotation has at least one pair of completely [[orthogonal]] planes A and B through O that are invariant as a whole, i.e. rotated in themselves.

[[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 a helical path on that torus. For a rotation whose two rotation angles have a rational ratio, 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.{{Sfn|Dorst|2019|pp=14−16|loc=6.2. Isoclinic Rotations in 4D}}]]

In general the rotation angles <math>\alpha</math> in plane A and <math>\beta</math> in plane B are different.

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 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>.

In that case of a double rotation, {{mvar|A}} and {{mvar|B}} are the only pair of invariant planes, and [[Ray (geometry)|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]] 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. 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.

====Isoclinic rotations====

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 perpendicalar 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.

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 [[Ray (geometry)|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-lines are invariant.{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}}

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.

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 OU to OX and from OY to OZ are reckoned positive. Then we have the four rotations R1 = (+<math>\alpha</math>, +<math>\alpha</math>), R2 = (-<math>\alpha</math>, -<math>\alpha</math>), R3 = (+<math>\alpha</math>, -<math>\alpha</math>) and R4 = (-<math>\alpha</math>, +<math>\alpha</math>). R1 and R2 are each other's [[inverse]]s; so are R3 and R4.

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''₁ {{=}} (+''α'', +''α'')}}, {{math|''R''₂ {{=}} (−''α'', −''α'')}}, {{math|''R''₃ {{=}} (+''α'', −''α'')}} and {{math|''R''₄ {{=}} (−''α'', +''α'')}}. {{math|''R''₁}} and {{math|''R''₂}} are each other's [[inverse function|inverses]]; so are {{math|''R''₃}} and {{math|''R''₄}}. 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''.

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>\alpha = 0</math> or <math>\alpha = \pi</math>.

<math>\alpha = 0</math> corresponds to the non-rotation; <math>\alpha = \pi</math> corresponds to the [[Inversion |central inversion]]. These two elements of SO(4) are the only ones which are left- ''and'' right-isoclinic.

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 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.

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]] 6-[[parameter]] [[Lie group]].

===Group structure of SO(4)===

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).

SO(4) is a [[noncommutative]] [[compact space|compact]] 6-[[dimension#Manifolds|dimensional]] [[Lie group]].

Each pairplane ofthrough completelythe [[orthogonal]]rotation planes throughcentre {{mvar|O}} is the pairaxis-plane of a [[invariantcommutative]] planes of a commutative [[subgroup]] of SO(4) [[isomorphic]] to SO(2)xSO(2). TheseAll these groupssubgroups are mutually [[maximal subgroup]]sconjugation of SO(4),isometries whichin are all mutuallyEuclidean 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)}}.

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).

These groups are [[maximal torus|maximal tori]] of SO(4), which are all mutually conjugate in SO(4). See also [[Clifford torus]].

Each left-isoclinic rotation [[commutative | commutes]] with each right-isoclinic rotation. This implies that there exists a [[direct product]] S3LxS3R with [[normal subgroup]]s S3L and S3R; both of the corresponding [[factor group]]s are isomorphic to the other factor of the direct product, i.e. isomorphic to S3.

All left-isoclinic rotations form a noncommutative subgroup {{math|''S''³_L}} of SO(4), which is isomorphic to the [[multiplicative group]] {{math|''S''³}} of unit [[quaternion]]s. All right-isoclinic rotations likewise form a subgroup {{math|''S''³_R}} of SO(4) isomorphic to {{math|''S''³}}. Both {{math|''S''³_L}} and {{math|''S''³_R}} are maximal subgroups of SO(4).

Each 4D rotation R is in two ways the product of left- and righ-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 [[Inversion |central inversion]] their product is R again.

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''³_L × ''S''³_R}}}} with [[normal subgroup]]s {{math|''S''³_L}} and {{math|''S''³_R}}; both of the corresponding [[factor group]]s are isomorphic to the other factor of the direct product, i.e. isomorphic to {{math|''S''³}}. (This is not SO(4) or a subgroup of it, because {{math|''S''³_L}} and {{math|''S''³_R}} are not disjoint: the identity {{mvar|I}} and the central inversion {{math|−''I''}} each belong to both {{math|''S''³_L}} and {{math|''S''³_R}}.)

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).

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

=== Special property of SO(4) among rotation groups in general ===

The odd-dimensional rotation groups do not contain the [[Inversion |central inversion]] and are [[simple group]]s.

This implies that {{math|''S''³_L × ''S''³_R}} is the [[universal covering group]] of SO(4) — its unique [[double covering group|double cover]] — and that {{math|''S''³_L}} and {{math|''S''³_R}} are normal subgroups of SO(4). The identity rotation {{mvar|I}} and the central inversion {{math|−''I''}} form a group {{math|C₂}} of order 2, which is the [[center of a group|centre]] of SO(4) and of both {{math|''S''³_L}} and {{math|''S''³_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)&nbsp;×&nbsp;SO(3). The factor groups of {{math|''S''}}³_L by C₂ and of {{math|''S''}}³_R by C₂ are each isomorphic to SO(3). Similarly, the factor groups of SO(4) by {{math|''S''}}³_L and of SO(4) by {{math|''S''}}³_R are each isomorphic to SO(3).

The even-dimensional rotation groups do contain the central inversion -I and have the group C2 = {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 centre are simple groups.

The topology of SO(4) is the same as that of the Lie group {{nowrap|1=SO(3) × Spin(3) = SO(3) × SU(2)}}, namely the space <math>\mathbb{P}^3 \times \mathbb{S}^3</math> where <math>\mathbb{P}^3</math> is the [[real projective space]] of dimension 3 and <math>\mathbb{S}^3</math> is the [[3-sphere]]. 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)}}.

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.

===Special property of SO(4) among rotation groups in general===

== Algebra of 4D rotations ==

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=C₂ = <nowiki>{</nowiki>{{math|''I''}}, {{math|−''I''}}<nowiki>}</nowiki>}} as their [[center of a group|centre]]. For even n ≥ 6, SO(n) is almost simple in that the [[factor group]] SO(n)/C₂ of SO(n) by its centre is a simple group.

SO(4) is commonly identified with the group of [[orientation]]-preserving [[Isometry|isometric]] [[linear]] mappings of a 4D [[vector space]] with [[inner product]] over the [[reals]] onto itself.

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 distinct subgroups {{math|''S''³_L}} and {{math|''S''³_R}} are conjugate to each other, and so cannot be 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 two isoclinic rotations through the same angle are conjugate. The set of all isoclinic rotations is not even a subgroup of SO(2{{math|''N''}}), let alone a normal subgroup.

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.

==Algebra of 4D rotations==

=== Isoclinic decomposition ===

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.{{Sfn|Kim|Rote|2016|loc=§5 Four Dimensional Rotations}}

A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation as follows:

===Isoclinic decomposition===

A 4D rotation given by its matrix is decomposed into a left-isoclinic and a right-isoclinic rotation<ref>{{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}}</ref> as follows:

Let

:<math>A=

\begin{pmatrix}

a_{00} & a_{01} & a_{02} & a_{03} \\

Line 75 ⟶ 78:

\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}

Line 88 ⟶ 91:

</math>

{{mvar|M}} has [[Rank_rank (linear_algebra)linear algebra)| rank]] one and is of unit [[Norm_(mathematics) | Euclidean norm]] as a 16D vector if and only if {{mvar|A}} is indeed a 4D [[rotation matrix]]. In this case there exist realsreal 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

and <math>(ap)^2 + ... + (ds)^2 = </math><math>(a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) = 1</math>.

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

There are exactly two sets of {{math|''a'', ''b'', ''c'', ''d''}} and {{math|''p'', ''q'', ''r'', ''s''}} such that {{math|''a''² + ''b''² + ''c''² + ''d''² {{=}} 1}} and {{math|''p''² + ''q''² + ''r''² + ''s''² {{=}} 1}}. They are each other's opposites.

The rotation matrix then equals

:<math>\begin{align}A&=

<math>A=

\begin{pmatrix}

ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\

Line 110 ⟶ 113:

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\\

Line 120:

d&-c&\;\,\, b&\;\,\, a

\end{pmatrix}

.

\begin{pmatrix}

p&-q&-r&-s\\

Line 127 ⟶ 126:

s&\;\,\, r&-q&\;\,\, p

\end{pmatrix}

.\end{align}

.

</math>

Line 134 ⟶ 133:

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 4D4-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 QL{{math|1=''Q''_L = ''a'' + ''bi'' + ''cj'' + ''dk''}}. In matrix-vector language this is

:<math>

<math>

\begin{pmatrix}

u'\\x'\\y'\\z'

Line 150 ⟶ 148:

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 QR{{math|1=''Q''_R = ''p'' + ''qi'' + ''rj'' + ''sk''}}, which is in matrix-vector form

:<math>

<math>

\begin{pmatrix}

u'\\x'\\y'\\z'

Line 169 ⟶ 165:

s&\;\,\, r&-q&\;\,\, p

\end{pmatrix}

.

\begin{pmatrix}

u\\x\\y\\z

\end{pmatrix}.

</math>

In the preceding section ([[#Isoclinic decomposition|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), or in symbolic form P' = QL.P.QR.</math>

or, in symbolic form,

Quaternion multiplication is [[associative]]. Therefore P' = (QL.P).QR = QL.(P.QR), which shows that left-isoclinic and right-isoclinic rotations commute.

:<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.<ref>{{cite journal

=== The Euler-Rodrigues formula for 3D rotations ===

| last1 = Rao | first1 = Dhvanita R.

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

| last2 = Kolte | first2 = Sagar

| doi = 10.1016/j.jalgebra.2018.05.026

| journal = Journal of Algebra

| mr = 3828791

| pages = 458–468

| title = Odd orthogonal matrices and the non-injectivity of the Vaserstein symbol

| volume = 510

| year = 2018}}</ref>

Quaternion multiplication is [[associative]]. Therefore,

<math>

:<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''_L}} and {{math|''Q''_R}} 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''_L}} and {{math|''Q''_R}} 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 p = a, q = -b, r = -c, s = -d, or in quaternion representation: QR = QL' = QL^(-1).

The 3D rotation matrix then becomes

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''_''R'' {{=}} ''Q''_''L''′ {{=}} ''Q''_''L''⁻¹}}.

<math>

The 3D rotation matrix then becomes the [[Euler–Rodrigues formula]] for 3D rotations

:<math>

\begin{pmatrix}

a_{11} & a_{12} & a_{13} \\

a_{21} & a_{22} & a_{23} \\

a_{31} & a_{32} & a_{33} \\

\end{pmatrix}

=

Line 206 ⟶ 219:

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 |Euler-RodriguesEuler–Rodrigues parameters]]: {{math|''a'', ''b'', ''c'', ''d''}}.

The corresponding quaternion formula {{math|''P′'' {{=}} ''QPQ''⁻¹}}, where {{math|''Q'' {{=}} ''Q''_L}}, or, in expanded form:

The corresponding quaternion formula P' = Q.P.Q^(-1) 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 [[William_Rowan_Hamilton |Hamilton]]-[[Arthur_Cayley |Cayley]] formula.

:<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=Hopf alsocoordinates===

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''² + ''y''² + ''z''² {{=}} 1}}, the points (''x'',''y'',''z'') lie on the unit 2-sphere. A point with angles {{math|{''θ''₀, ''φ''₀<nowiki>}</nowiki>}}, rotated by an angle {{mvar|φ}} about the {{mvar|z}}-axis, becomes the point with angles {{math|{''θ''₀, ''φ''₀ + ''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 [[Hopf coordinates]] {{math|{''ξ''₁, ''η'', ''ξ''₂<nowiki>}</nowiki>}},<ref name="Karcher">{{Citation

*[[orthogonal matrix]]

|url= http://virtualmathmuseum.org/Surface/bianchi-pinkall_tori/bianchi-pinkall_tori.html

*[[orthogonal group]]

|title= Bianchi–Pinkall Flat Tori in S₃

*[[rotation group]]

|work= 3DXM Documentation

*[[Lorentz group]]

|last= Karcher |first= Hermann

|publisher= 3DXM Consortium

|access-date= 5 April 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''² + ''x''² + ''y''² + ''z''² {{=}} 1}}, the points lie on the 3-sphere.

== References ==

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''₁}} and {{math|''ξ''₂}}. 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''₁₀, ''η''₀, ''ξ''₂₀<nowiki>}</nowiki>}} through angles {{math|''ξ''₁}} and {{math|''ξ''₂}} is then simply expressed in Hopf coordinates as {{math|{''ξ''₁₀ + ''ξ''₁, ''η''₀, ''ξ''₂₀ + ''ξ''₂<nowiki>}</nowiki>}}.

L. van Elfrinkhof: Eene eigenschap van de orthogonale substitutie van de vierde orde. ''Handelingen van het 6e Nederlandsch Natuurkundig en Geneeskundig Congres, Delft, 1897.

==Visualization of 4D rotations==

[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.

[[File:4DRotationTrajectories.jpg|thumb|upright=1.75|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 a fixed axis unchanged by 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''₀, ''φ''₀<nowiki>}</nowiki>}} on the sphere, under a rotation about the {{mvar|z}}-axis, will follow a trajectory {{math|{''θ''₀, ''φ''₀ + ''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''₁, ''η'', ''ξ''₂<nowiki>}</nowiki>}} parameterize the 3-sphere. For fixed {{mvar|η}} they describe a torus parameterized by {{math|''ξ''₁}} and {{math|''ξ''₂}}, 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''₁₀, ''η''₀, ''ξ''₂₀<nowiki>}</nowiki>}} undergoing a rotation with the {{mvar|uz}}- and {{mvar|xy}}-planes invariant will remain on the torus specified by {{math|''η''₀}}.<ref name="Pinkall">{{cite journal |last=Pinkall |first=U. |date=1985 |title=Hopf tori in S₃ |url=https://www.maths.ed.ac.uk/~aar/papers/pinkall.pdf |journal=Invent. Math. |volume=81 |issue=2 |pages=379–386 |access-date=7 April 2015 |doi=10.1007/bf01389060|bibcode=1985InMat..81..379P |s2cid=120226082 }}</ref> The trajectory of a point can be written as a function of time as {{math|{''ξ''₁₀ + ''ω''₁''t'', ''η''₀, ''ξ''₂₀ + ''ω''₂''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 |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''₁ {{=}} 1}} and {{math|''ω''₂ {{=}} 5}} is shown, while in Fig. 3, a general rotation in which {{math|''ω''₁ {{=}} 5}} and {{math|''ω''₂ {{=}} 1}} is shown.

Below, a spinning [[5-cell]] is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus described above is depicted in its rectangular (wrapping) form.

<gallery caption="Animated 4D rotations of a [[5-cell]] in [[orthographic projection]]">

File:Simple 4D rotation of a 5-cell, in X-Y plane.webm|loop|Simply rotating in X-Y plane

File:Simple 4D rotation of a 5-cell, in Z-W plane.webm|loop|Simply rotating in Z-W plane

File:Double 4D rotation of a 5-cell.webm|loop|Double rotating in X-Y and Z-W planes with angular velocities in a 4:3 ratio

File:Isoclinic left 4D rotation of a 5-cell.webm|loop|Left isoclinic rotation

File:Isoclinic right 4D rotation of a 5-cell.webm|loop|Right isoclinic rotation

</gallery>

==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''₁}} and {{math|''A''₂}} satisfying the properties {{math|''A''₁''A''₂ {{=}} 0}}, {{math|''A''₁³ {{=}} −''A''₁}} and {{math|''A''₂³ {{=}} −''A''₂}}, where {{math|∓''θ''₁''i''}} and {{math|∓''θ''₂''i''}} are the eigenvalues of {{mvar|A}}. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices {{math|''A''₁}} and {{math|''A''₂}} by Rodrigues' rotation formula and the Cayley formula.<ref>{{cite web |last1=Erdoğdu |first1=M. |last2=Özdemir |first2=M. |date=2015 |title=Generating Four Dimensional Rotation Matrices |url=https://www.researchgate.net/publication/283007638}}</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''₁}} and {{math|''A''₂}} 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''₁}} and {{math|''A''₂}} 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'''⁴}}, 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'''⁴}}, 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''₁}} and {{math|''θ''₂}} as follows:

# If {{math|''θ''₁ {{=}} 0}} and {{math|''θ''₂ ≠ 0}} or vice versa, then the formulae generate simple rotations;

# If {{math|''θ''₁}} and {{math|''θ''₂}} are nonzero and {{math|''θ''₁ ≠ ''θ''₂}}, then the formulae generate double rotations;

# If {{math|''θ''₁}} and {{math|''θ''₂}} are nonzero and {{math|''θ''₁ {{=}} ''θ''₂}}, 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==

In this monography 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 [[Hilbert]] to four dimensions.

{{Notelist|group=nb}}

==References==

[http://www.arxiv.org/PS_cache/math/pdf/0501/0501249.pdf Johan E. Mebius: ''A matrix-based proof of the quaternion representation theorem for four-dimensional rotations.''] WWW: arXiv General Mathematics 2005.

{{Reflist}}

==Bibliography==

[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.

*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.

*[https://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.

*{{cite journal |jstor=1986315 |title=Quaternion Space |last1=Hathaway |first1=Arthur S. |author-link=Arthur Stafford Hathaway |journal=[[Transactions of the American Mathematical Society]] |year=1902 |volume=3 |issue=1 |pages=46–59 |doi=10.1090/S0002-9947-1902-1500586-2|doi-access=free }}

*{{cite arXiv |eprint=math/0501249 |author1=Johan Ernest Mebius |title=A matrix-based proof of the quaternion representation theorem for four-dimensional rotations |year=2005}}

*{{cite arXiv |eprint=math/0701759 |author1=Johan Ernest Mebius |title=Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations |year=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 |author-link= 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= 2 |pages= 183–214 |jstor=1986218 |doi= 10.1090/s0002-9947-1901-1500564-2 |doi-access= free }}

*{{cite journal |doi=10.36753/mathenot.642208 |title=Simple, Double and Isoclinic Rotations with Applications |year=2020 |last1=Erdoğdu |first1=Melek |last2=Özdemi̇r |first2=Mustafa |journal=Mathematical Sciences and Applications E-Notes|doi-access=free }}

*{{cite journal |first=Daniele |last=Mortari |title=On the Rigid Rotation Concept in n-Dimensional Spaces |url=https://pdfs.semanticscholar.org/f7d8/63ceb75277133592ef9e92457b6705b1264f.pdf |journal=Journal of the Astronautical Sciences |volume=49 |issue=3 |date=July 2001 |pages=401–420 |doi=10.1007/BF03546230 |bibcode=2001JAnSc..49..401M |s2cid=16952309 |archive-url=https://web.archive.org/web/20190217142347/https://pdfs.semanticscholar.org/f7d8/63ceb75277133592ef9e92457b6705b1264f.pdf |archive-date=2019-02-17 }}

* {{cite arXiv |last1=Kim |first1=Heuna |last2=Rote |first2=G. |date=2016 |title=Congruence Testing of Point Sets in 4 Dimensions |class=cs.CG |eprint=1603.07269}}

*{{cite journal | arxiv=2003.09236 | date=8 Jan 2021 | last=Zamboj | first=Michal | title=Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space | journal=Journal of Computational Design and Engineering | volume=8 | issue=3 | pages=836–854 | doi=10.1093/jcde/qwab018 }}

* {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159|doi-access=free}}

[[Category:LieFour-dimensional groupsgeometry]]

[[Category:Quaternions]]

[[Category:4-dimensional geometryRotation]]