We talk about the earth rotating on its axis. We say the same thing about tops, and spinning basketballs, and gyroscopes, and car tires, and pretty much everything else that spins. Rotations happen around an axis.
Well, except that they actually don't.
No, I'm serious. Rotations happen in a plane, and the fact that a plane happens to correspond to a unique perpendicular axis is just a lucky coincidence that occurs in three dimensional space and nowhere else. The earth rotates in the plane of its equator. If you want to make a rotation in 4-d or higher space, you'll actually need to specify the plane of rotation, as there isn't a unique axis perpendicular to the plane or rotation.
What difference does this make since there are in fact only three dimensions? It makes a difference in the abstract 4-dimensional space of relativity where we count time as a dimension on a (mostly) equal footing with the three spatial dimensions. Changing from one frame of reference to another can be done in terms of rotations of that 4-d space. There are more speculative theories which involve spaces with a much higher number of dimensions. 10, 11, and 26 frequently appear. In those cases you have to be precise about exactly what you mean by a rotation.
To everything, turn, turn, turn. Just do it in a plane, not about an axis.
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In 3D, there is one axis of rotation. There is an infinite number of planes in which things rotate -- the equator is only ONE of them. I think you are confusing the terms "rotate about" (or "rotate around") with "rotate in". In 3D, the rotation is about an axis. One particular point undergoing such a rotation will have its path in a plane perpendicular to that axis, but other points may have their rotations in different planes. For example, a person on the equator and a person on the 45th parallel will rotate about the same axis as the Earth turns, but they rotate in different planes. An axis uniquely determines rotation of 3D objects in 3D, but a plane does not.
Rotating about a plane is also a special case to 4D. In 2D, you rotate about a point. In 3D, about a line. In 4D, about a plane. In 5D, about a 3-volume. In 6D, about a 4-volume. Etc, etc. Rotation is always about something 2 dimensions smaller than the number of dimensions of the space you are in. This is because, in N-dimensional space, a rotation corresponds to a change in a single angle for a suitable choice of coordinate system, which changes only 2 coordinates while leaving N-2 fixed.
To elaborate on my last point, in N-dimensional space, we can always choose a coordinate system such that an arbitrary point x = (x1,x2,x3,...,xN) under rotation through an angle theta goes to x' = (x1',x2',x3',...,xN') where:
x1' = x1 cos(theta) - x2 sin(theta)
x2' = x1 sin(theta) + x2 cos(theta)
x3' = x3
x4' = x4
...
xN' = xN
for any point x.
@CS: I think your first paragraph overcomplicates things. When Matt speaks of rotation in a plane, he seems to implicitly include all planes parallel to the specified plane, as the mathematics in your follow-up post makes clear.
Additional remark: The reasoning given in the post also explains why computing the cross product of two vectors as a third vector* only makes sense in three dimensions. What you are really doing in computing a cross product is constructing an antisymmetric tensor from the outer product of the vectors. For example, if a = (a1, a2, a3, ...) and b = (b1, b2, b3, ...), then a X b yields (up to a sign convention) (and hoping this comes out halfway legible without HTML table formatting)
0 a2b1-a1b2 a3b1-a1b3 ...
a1b2-a2b1 0 a3b2-a2b3 ...
a1b3-a3b1 a2b3-a3b2 0 ...
... ... ... ...
In N dimensions you have N(N-1)/2 unique elements in this tensor, so if N=3 you can identify these components with a vector: c1 = a2b3-a3b2, etc. For N>3 you are stuck with the tensor.
*Technically, the cross product in 3D is a pseudovector. The difference: Coordinate inversion (x,y,z)->(-x,-y,-z) will map a vector a into -a, but if c is the cross product of two vectors, coordinate inversion will map c into itself.
Re rotations and all that. Matt, you might print out David Hestenes' Oersted Medal Lecture on geometry and put it where you can read it in one of those all too brief spare moments that happen to grad students. You can think of it as the geometric foundation of Dirac's gamma matrices.
Exactly right, CS and Eric. What I mean is that as per Eric's generalization in comment #2, we define a rotation with only the x1 and x2 coordinates, which specify a plane and (as mentioned) implicitly the planes parallel to that. That we can construct a unique orthogonal vector from such a transformation is just happenstance in 3D. My "rotate about" and "rotate in" are of necessity a little vague given that we're working from what amounts to visual analogy in N dimensional space.
Eric Lund is half-right. [Partial credit shown here in red ink]. "In N dimensions you have N(N-1)/2 unique elements in this tensor, so if N=3 you can identify these components with a vector: c1 = a2b3-a3b2, etc. For N>3 you are stuck with the tensor."
Suggestion: look more carefully at the N=7 case, and hand in your homework later. I won't take away points for the extra time.
I am typing this on a "staff use only" PC at Nia Educational Charter School where I'm the full-time science teacher, and my mind is reeling from grading homeworks and quizzes. Reeling. Rotating. Which is where we came in...
Classical physics has never handled odd (ungerade for chemists) functions well - pseudoscalars, polar vectors, pseudotensors; rotation, angular momentum, chirality; Green's function vs. GR and GR vs. teleparallelism. If quantum folk are feeling superior at this point, Yang and Lee.
The vacuum is isotropic because angular momentum is conserved (Noether's theorem). Chiral vacuum is the 30.5 A diameter hollow inside of point group I (not Ih) C980 fullerene. Insert [R]- or [S]- point group T (not Th or Td) C21. Starting with coincident centers of mass they fall and spin divergently.
Nobody has ever tested physics' founding postulates with chemically identical opposite parity mas distributions. There is every reason to believe a parity Eotvos experimtn opposing single crystal enantiomorphic space groups P3(1)21 adn P3(2)21 quartz test masses will do a Yang and Lee on gravitation thence conservation of angular momentum.
Closely related is that the vector product axb is only defined in 3D, but it can be generalized by defining the "orientation" of the result as a "directed area". See
http://en.wikipedia.org/wiki/Wedge_product