[rmidthun]

Rotations can be composed in up to 3 dimensions only. That means that for any sequence of rotations, you can come up with a single rotation that is equal to that rotation. In 2d, this is just adding scalars and rotation has no meaning in 1 or 0 dimensions.

In 4 dimensions, this is not true. A rotation in XY cannot compose with one in ZW so at least two rotations are required.

[pandaman]

Could you explain more about 4D rotation? In my understanding, a rotation is a transform by an orthogonal matrix with determinant equal to 1. For the 4D rotations to be not composable the product of two 4x4 orthogonal matrices should not be an orthogonal matrix. Is this really true?

[abecedarius]

By 'rotation' is meant rotating a plane that goes through the origin, carrying along the other points that are off the plane according to their projection onto it. In 4-d if you rotate around the y-z plane, then around the w-x plane, there's no plane that can express a single composed rotation with the same effect. But as you say, it's composable as a unitary matrix.

[pandaman]

Hmmm, I am still confused. Rotating "around plane" makes sense only in 3D, where a 2D plane is a hyperplane i.e. has a single normal and you can rotate around that normal (which you implied by mentioning projections along that normal). In 4D a hyperplane is a 3D subspace, orthogonal to some normal. So, while it may be true that you cannot find a 2D plane describing a composite rotation in 4D it does not seem to have any significance since "rotation around y-z" plane does not describe one either :) because y-z plane has an infinite number of normals in 4D and any rotation around any of them will be technically a rotation "around y-z plane" while a different rotation in 4D. I am sure you can find a 3D hyperplane in 4D for any composite rotation. Am I wrong?

[abecedarius]

You're quite right that there's no unique normal to a 2-d plane in 4-d, and so it's meaningless to "rotate around the normal vector". That's why I tried to phrase the geometry in terms of planes instead of axes.

Here's an example that's easiest to sort-of-visualize:

1. Rotate the y-z plane of 4-d space by half a turn (180 degrees). That is, transform every point in 4-space according to w->w, x->x, y->-y, z->-z.

2. Rotate the w-x plane, also a half-turn. So w->-w, x->-x, y->y, z->z.

Composing these two transformations yields w->-w, x->-x, y->-y, z->-z. That composed transformation is an inversion, not a rotation. Is it clearer now? (It was late when I wrote last night, sorry. Let me try and rephrase how I defined a rotation: a linear transformation that carries one unit vector to a second unit vector, and leaves unchanged any vectors that are orthogonal to both. In my previous comment the plane I was talking about was the plane that contains both given vectors. Of course, the null rotation doesn't pick out any particular plane, though it's not otherwise special in these terms. IANAM, but would appreciate correction if I'm goofing.)

[pandaman]

I think I see what you mean. In an even dimensions space a -x is a rotation ( determinant = 1), while it is not in an odd dimensions. So what you described is a rotation in 4D but, indeed, not in 3D. Also, I feel, from this follows, that you can only have a rotation axis (eigenvector for value 1) in an odd dimensions space so I was wrong, assuming there is an axis for any 4D rotation. Same as in 2D there may be none. I figure 5D will act as same as 3D in this regard.

[abecedarius]

I'm sorry, my 'rotation' was wrong -- as you point out, it would admit reflections. I'm glad to uncover this embarrassing bug, but on the other hand, oops. https://en.wikipedia.org/wiki/Rotation_(mathematics) has more lore to check against after I've done enough mental debugging.