Timeline for Symmetrical Presentation of 4-Dimensional Rotation Matrix
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 15, 2009 at 18:37 | vote | accept | Rhubbarb | ||
| Nov 11, 2009 at 14:39 | comment | added | Danny Calegari | You can think of a pair of complex numbers (ie a vector in $C^2$) as a 4-tuple of real numbers (by taking the real and imaginary part of each complex number). So a transformation which multiplies each complex number by $e^{i\theta}$ "is" a rotation of (real) 4-space in which every vector rotates (through the same angle $\theta$). | |
| Nov 11, 2009 at 9:43 | comment | added | Rhubbarb | DC: Your reply did highlight that I'd made a false assumption: that 4D rotations must have fixed points, as do 3D rotations. However, there is at least one simple counter example: the map (w,x,y,z) --> (-w,-x,-y,-z). Because the dimension is even, this is a true rigid rotation, and does not have a reflection component. This particular map may be thought of as some kind of inversion or reflection through a point. | |
| Nov 11, 2009 at 9:39 | comment | added | Rhubbarb | DC: Thanks; could you please briefly explain multiplication by e^(i theta)? I understand this in 2D, as it applies to complex numbers (pairs of reals) where this may also be written cis(theta) or cos(theta)+i*sin(theta), and thus corresponds to a the 2D matrix [[c,s],[-s,c]]. How is this extended to 4-tuples please? | |
| Nov 10, 2009 at 22:15 | history | answered | Danny Calegari | CC BY-SA 2.5 |