Talk:Rotations in 4-dimensional Euclidean space

Central Inversion

The link central inversion in the Group structure of SO(4) section is to nothing useful. Someone please find or stub a relevant page to link to instead. —Preceding unsigned comment added by 76.119.66.12 (talk) 23:25, 22 January 2009 (UTC)

Relation to orbital mechanics

The article on the Laplace-Runge-Lenz vector has the set of commutation relations for the Lie algebra of SO(4) as they are commonly taught to physics students. These are notably absent from this article. These are important for several reasons: (1) they help emphasize that the difference between this and SO(3,1) is a change of sign, and (2) they help emphasize the decomposition of of SO(3,1) as the product of a pair of complex conjugate reps of SL(2,C). I'm not clear on what the decomposition of the adjoint rep of SO(4) is w.r.t. the fundamental rep. In particular, I'd like to see a deeper treatment of the representations of SO(4), and the homotopy groups, etc. linas 02:50, 18 June 2006 (UTC)

How can two planes intersect in a point?

"Every plane B that is completely orthogonal (*) to A intersects A in a certain point P." As far as I know planes intersect in planes, lines or not at all.

That's true only in three dimensions. In four dimensions, planes generically intersect in points. Take for example the plane spanned by the first two coordinate vectors and the plane spanned by the last two. These two planes intersect only at the origin. -- Fropuff 16:30, 10 November 2006 (UTC)pla

The previous paragraph refers to 2-planes (2-dimensional planes) in 4-space. For instance, the xy coordinate plane in 4-space is {(x,y,0,0)}, and the zw-coordinate plane is {(0,0,z,w)} (where x, y, z, w each independently range over all real numbers). Clearly the only point in common to the two 2-planes is (0,0,0,0). For arbitrary planes in 4-space — each having dimension 0, 1, 2, 3, or 4 — the intersection of a p-plane and a q-plane is usually a (p+q-4)-plane. Here a 0-plane means just a single point, and a "plane" with negative dimension just means the empty set. — Preceding unsigned comment added by 50.205.142.35 (talk • contribs) 15:00, 17 January 2020 (UTC)

Please sign all your talk page messages with four tildes (~~~~) — See Help:Using talk pages. Thanks. - DVdm (talk) 15:35, 17 January 2020 (UTC)

Dead Links

The two links to Johan E. Mebius on ArXiv.org no longer exist. Is there an alternative location?

Location found and links fixed.

Proposed move

The title is far to obscure. I propose that it should be moved to another less obscure title. Hexadecachoron ^talk _contribs 10:42, 16 September 2009 (UTC)

I agree.

I have moved to page from SO(4) to Rotations in 4-dimensional Euclidean space. I hope this is acceptable. Jheald (talk) 13:16, 27 July 2011 (UTC)

Maybe SO(4) is obscure (for the uninitiated), but "Rotations in 4-dimensional Euclidean space" is confusing, since for most people rotations are about an axis and by an angle, what is called a simple rotation in this article. If you don't care about axes and angles, the unambiguous name (for an element of SO(n) in general) would be "direct orthogonal transformation". Marc van Leeuwen (talk) 15:30, 21 August 2011 (UTC)

A non-simple rotation can still be decomposed as a (simple) rotation followed by another rotation, so I don't see that the new title is so misleading. And even the non-simple rotations are explicitly called "double rotations". Jheald (talk) 17:55, 21 August 2011 (UTC)

A composition of rotations is not the same thing as a rotation, just like a composition of reflections is not the same thing as a reflection. For me the statement that in 3D the composition of two rotations fixing the origin is again a rotation is a non-trivial theorem (basically proved by a classification of elements of SO(3)). This can be seen from the fact that without "fixing the origin" (i.e., for affine rotations) it is false: the composition could be for instance a translation, which I think nobody would call a rotation. I think the term rotation is not widely used in dimensions 4 and higher (just look at Rotation and Rotation matrix), so in a sense anyone can define it the way she likes; however in WP we should adhere to established terminology (and provide sources to prove this). Personally, I find the reflection/rotation similarity appealing, and would, should I have to define rotations in any dimension, stick to the same what-it-does description: fix a hyperplane and act by -1 on the orthogonal complement for reflections, fix a codimension 2 space and act by a rotation on the orthogonal complement for rotations. Also just as reflections are the simplest generating set for O(n) (or its affine counterpart), so are the reflections for SO(n) (and just like transposition respectively 3-cycles do for the symmetric and alternating groups). Marc van Leeuwen (talk) 12:20, 22 August 2011 (UTC)

I agree the name now is accurate. It's very long though, for something most mathematicians would simply call "4D rotations" or "rotations in 4D" – Euclidean space, or at least Euclidean, can usually be assumed. Compare e.g. Four-dimensional space or Rotation around a fixed axis. Perhaps loose the last two words and make it Rotations in 4 dimensions ?--JohnBlackburne^words_deeds 22:54, 21 August 2011 (UTC)

For a physicist 4D space is most usually going to refer to Minkowski space. Lorentz boosts in such a space are often called rotations; but they aren't the kind of rotations the article is dealing with here. So that's why I thought it was appropriate to make clear in the title that this article was to do with rotations specifically in four Euclidean dimensions. Yes, that makes for quite a long title; but it seems to me quite a natural phrase, and as most people will be following links to get here, I'm not sure the length of the title is really very much to worry about.

On the other hand, redirects are cheap and it would make a lot of sense to have both 4D rotations and Rotations in 4D as redirects forwarding to here. But I think there may be advantages in keeping the longer more explicit phrase as the canonical title for the article. Jheald (talk) 11:30, 22 August 2011 (UTC)

The word "rotations" is precisely the correct technical term for elements of SO(4) acting on 4-dimensional euclidean space. A composition of rotations is absolutely a rotation. The term "rotation" is used All. The. Time. in dimensions 4 and higher (as well as lower dimensions). Ignorance is not bliss.

Double rotations

In the section on double rotations, it is written that "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." To me, this is not clear. I believe that the meaning is that each of the two planes is invariant as a whole (but I have no precise source on this). However, this could be interpreted as the union of the two planes being invariant as a whole, which is a weaker statement. Can anyone please reformulate? (I guess the best source to check is the article by van Elfrinkhof, if you understand Dutch, or any of the sources cited there.) 130.238.58.123 (talk) 14:32, 16 January 2012 (UTC) I have now seen that the article on SO(n) is more precise on this subject, and could be cited instead.

Error

There is an error in the math code. Someone should fix it! — Preceding unsigned comment added by 65.128.190.136 (talk) 16:16, 5 October 2012 (UTC)

Error

There is an error in the math code. Someone should fix it! — Preceding unsigned comment added by 65.128.190.136 (talk) 16:16, 5 October 2012 (UTC)

Problem with single rotation

I'm having a problem with a single rotation. Suppose I have a unit quaternion QL=[a,b,c,d]. In the article it states "In quaternion notation, a rotation in SO(4) is a single rotation if and only if QL and QR are conjugate elements of the group of unit quaternions." So if I want to rotate a point at r=[0,0,0,1] (i.e. a point displaced 1 unit along the z axis), the rotated point will be:

$r'=Q_{L}\,r\,Q_{L}^{*}=[0,2ac+2bd,-2ab+2cd,a^{2}-b^{2}-c^{2}+d^{2}]$

which means that the point can never be rotated into the u dimension, it must be constrained to the xyz space. This seems wrong, one should be able to rotate [0,0,0,1] into the u dimension. PAR (talk) 03:50, 5 December 2014 (UTC)

This is not really the place for such mathematical problems. You should ask at the maths reference desk, WP:RD/MA, where you are more likely to get an answer (or often many answers).--JohnBlackburne^words_deeds 04:45, 5 December 2014 (UTC)

Ok, thanks, will do.PAR (talk) 16:46, 5 December 2014 (UTC)

Relation to quaternions

Are the two

${\begin{pmatrix}u'\\x'\\y'\\z'\end{pmatrix}},$

both left-multiplied, really the same? Moreover, the text with "right-multiplication" highly suggests:

${\begin{pmatrix}u'&x'&y'&z'\end{pmatrix}}={\begin{pmatrix}u&x&y&z\end{pmatrix}}\cdot {\begin{pmatrix}p&-q&-r&-s\\q&\;\,\,p&\;\,\,s&-r\\r&-s&\;\,\,p&\;\,\,q\\s&\;\,\,r&-q&\;\,\,p\end{pmatrix}}$

Nomen4Omen (talk) 16:36, 18 June 2016 (UTC)

Apparent misstatement

"Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product S³_L × S³_R with normal subgroups S³_L and S³_R; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to S³. (This is not SO(4) or a subgroup of it, because S³_L and S³_R are not disjoint: the identity I and the central inversion −I each belong to both S³_L and S³_R.)"

The phrase "This is not SO(4) or a subgroup of it" seems to be saying that the group S³ is not isomorphic to a subgroup of SO(4). But this is contradicted by the previous sentence (which is correct).

If that is not what this phrase is meant to say, then it needs to be rewritten to express its intended meaning clearly.50.205.142.35 (talk) 14:48, 17 January 2020 (UTC)