Wow, I didn’t know that. It makes sense now I think about it though; SO(n) must be something like an n(n-1)/2 dimensional space, but the space of rotations about an (n-2)-subspace must be … err … something smaller—maybe 2n-3 dimensional? I may be abusing the idea of dimension here...
First of all, terminology. SO(n) is orientation-preserving orthogonal transformations on n-space, or equivalently the orientation-preserving symmetries of an (n-1)-sphere in n-space. So Joshua’s statement is about SO(n) for n>3.
OK. So the obvious way to interpret “rotation about an axis” in many dimensions is: you choose a 2-dimensional subspace V, then represent an arbitrary vector as v+w with v in V and w in its orthogonal complement, and then you rotate v. The dimension of the set of these things is (n-1)+(n-2) from choosing V—you can pick one unit vector to be in V, and then another unit vector orthogonal to it—plus 1 from choosing how far to rotate. So, 2n-2.
And yes, the dimension of SO(n) is n(n-1)/2. One way to see this: you’ve got matrices with n^2 elements, and n(n+1)/2 constraints on those elements because all the pairwise inner products of the columns (including each column with itself) are specified.
These dimensions are all topological dimensions rather than vector-space dimensions, since the sets we’re looking at aren’t vector subspaces of R^(n^2), but there’s nothing abusive about that :-).
It can’t be 2n-2 because it’s 3 when n=3. I get 2n-3 because the first vector is chosen with n-1 degrees of freedom, then the second with n-2, then subtract one because of the equivalence class of rotations, then add one for choosing how far to rotate.
EDIT: More generally, I think that the dimension of k-dimensional subspaces of an n-dimensional spaces is k(n-k), so where k=2 you get 2n-4, then add one for choosing how far to rotate. I’d feel better if I knew what I meant by “dimension” here though; it’s not a vector space.
As for topological dimension, roughly, if you consider a neighborhood of a point in the space, what does space look like from there? Locally it’s Euclidean if you’re “on” a manifold. The rigorous definition involves charts. See also Lebesgue covering dimension.
Meh, you’re right: the dimension of the space of 2-dimensional subspaces of n-space is 2n-4, not 2n-3. The reason why my handwavy dimension-counting above was wrong is (“of course”) that I failed to “subtract one because of the equivalence class of rotations”. And yes, you’re right that in general it’s k(n-k).
“Dimension” here means: locally the set looks like a that-many-dimensional vector space. That is, e.g., any element of SO(n) has a neighbourhood that’s topologically the same as a neighbourhood in R^(n(n-1)/2).
I’d feel better if I knew what I meant by “dimension” here though; it’s not a vector space.
The number of parameters you need to label each element (provided the labelling is a continuous function, otherwise you can label points of R^2 with a single parameter e.g. (3.1415..., 2.7182...) → 32.174118...)
It is certainly in use, I don’t know how widespread it is. Generally in high dimensions one is just interested in SO(n) anyways, so there’s not that much need to make the distinction in most contexts.
In this context rotations are rotations about some n-subspace by some angle rather than all oritentation preserving isometries.
Wow, I didn’t know that. It makes sense now I think about it though; SO(n) must be something like an n(n-1)/2 dimensional space, but the space of rotations about an (n-2)-subspace must be … err … something smaller—maybe 2n-3 dimensional? I may be abusing the idea of dimension here...
First of all, terminology. SO(n) is orientation-preserving orthogonal transformations on n-space, or equivalently the orientation-preserving symmetries of an (n-1)-sphere in n-space. So Joshua’s statement is about SO(n) for n>3.
OK. So the obvious way to interpret “rotation about an axis” in many dimensions is: you choose a 2-dimensional subspace V, then represent an arbitrary vector as v+w with v in V and w in its orthogonal complement, and then you rotate v. The dimension of the set of these things is (n-1)+(n-2) from choosing V—you can pick one unit vector to be in V, and then another unit vector orthogonal to it—plus 1 from choosing how far to rotate. So, 2n-2.
And yes, the dimension of SO(n) is n(n-1)/2. One way to see this: you’ve got matrices with n^2 elements, and n(n+1)/2 constraints on those elements because all the pairwise inner products of the columns (including each column with itself) are specified.
These dimensions are all topological dimensions rather than vector-space dimensions, since the sets we’re looking at aren’t vector subspaces of R^(n^2), but there’s nothing abusive about that :-).
It can’t be 2n-2 because it’s 3 when n=3. I get 2n-3 because the first vector is chosen with n-1 degrees of freedom, then the second with n-2, then subtract one because of the equivalence class of rotations, then add one for choosing how far to rotate.
EDIT: More generally, I think that the dimension of k-dimensional subspaces of an n-dimensional spaces is k(n-k), so where k=2 you get 2n-4, then add one for choosing how far to rotate. I’d feel better if I knew what I meant by “dimension” here though; it’s not a vector space.
These are the best references I know:
Calculus on Manifolds
Boothby
As for topological dimension, roughly, if you consider a neighborhood of a point in the space, what does space look like from there? Locally it’s Euclidean if you’re “on” a manifold. The rigorous definition involves charts. See also Lebesgue covering dimension.
Meh, you’re right: the dimension of the space of 2-dimensional subspaces of n-space is 2n-4, not 2n-3. The reason why my handwavy dimension-counting above was wrong is (“of course”) that I failed to “subtract one because of the equivalence class of rotations”. And yes, you’re right that in general it’s k(n-k).
“Dimension” here means: locally the set looks like a that-many-dimensional vector space. That is, e.g., any element of SO(n) has a neighbourhood that’s topologically the same as a neighbourhood in R^(n(n-1)/2).
This is correct.
The number of parameters you need to label each element (provided the labelling is a continuous function, otherwise you can label points of R^2 with a single parameter e.g. (3.1415..., 2.7182...) → 32.174118...)
To make this precise, you need the idea of “charts” and “atlases” that witzvo references.
I don’t recall encountering this usage before. Is it widespread?
It is certainly in use, I don’t know how widespread it is. Generally in high dimensions one is just interested in SO(n) anyways, so there’s not that much need to make the distinction in most contexts.