Talk:Rotations in 4-dimensional Euclidean space

message: 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)
thread_title: Central Inversion

message: 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)
thread_title: Relation to orbital mechanics

message: "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)
replies: text: 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)
replies: text: Please sign all your talk page messages with four tildes (~~~~) — See Help:Using talk pages. Thanks. - DVdm (talk) 15:35, 17 January 2020 (UTC)
thread_title: How can two planes intersect in a point?

message: 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)
thread_title:  Double rotations 

message: 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)
thread_title:  Error 

message: 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)
thread_title:  Error 

message: 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, 2 a c + 2 b d, -2 a b + 2 c d, 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)
replies: text: 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).--JohnBlackburnewordsdeeds 04:45, 5 December 2014 (UTC)
replies: text: Ok, thanks, will do.PAR (talk) 16:46, 5 December 2014 (UTC)
thread_title:  Problem with single rotation 

message: 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)
thread_title:  Relation to quaternions 

message: "Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct product  with normal subgroups 3L and 3R; both of the corresponding factor groups are isomorphic to the other factor of the direct product, i.e. isomorphic to 3. (This is not SO(4) or a subgroup of it, because 3L and 3R are not disjoint: the identity  and the central inversion  each belong to both 3L and 3R.)"
The phrase "This is not SO(4) or a subgroup of it" seems to be saying that the group 3 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)
thread_title:  Apparent misstatement