# Rotations in 3 dimensions

I was trying to think of how to write a general rotation matrix in $\mathbb R^3$ but was struggling a bit. I thought that there would be two independent parameters, say an angle for rotating about the $\textbf{k}$ unit vector as in the 2D case, and then some other angle but I couldn't figure out what this other angle should be a rotation about. So I looked up rotation matrices and was surprised to find (both Wolfram alpha and Wikipedia say) that the rotation in $\mathbb R^3$ is a composition of a rotation about each of the axes with some independent angle. So that means three independent parameters?

Perhaps I am understanding something wrong here, but why is it the case that in a plane (2D) you need only one angle $\theta$ to describe a rotation and here in $\mathbb R^3$ it is saying that we need three?

• Does this help? This representation depends on three angles. math.stackexchange.com/questions/2395827/… – Randall Aug 27 '17 at 13:43
• Or you could say you need an axis, and an angle. The axis corresponds to a pair of antipodal points on the unit sphere, so has two degrees of freedom. – Lord Shark the Unknown Aug 27 '17 at 14:04
• Let's say you wanted to rotate the earth. You can pick an arbitary position for the new north pole. (That's 2 parameters.) Now you get to spin in about its new axis so that Paris can be at any point on its latitude. That's one more parameter. – kimchi lover Aug 27 '17 at 14:41

As it turns out, there is a (reasonably) simple relationship between the number of dimensions of an $n$-dimensional Euclidean space (such as $\mathbb R^n$ with a suitable metric) and the number of parameters required to specify a rotation in that space.
As explained here and in the comments of this question, in order to fully specify an arbitrary rotation in $n$ dimensions you need $n(n-1)/2$ parameters. For the first few dimensions, the number of parameters is shown in the table below. \begin{matrix} n && n(n-1)/2 \\ \hline 1 && 0 \\ 2 && 1 \\ 3 && 3 \\ 4 && 6 \end{matrix}
One way to count the number of parameters is to assign one parameter for each pair of axes in the $n$-dimensional space. This obviously works for one or two dimensions; for three dimensions it gives us the Euler angles; for four or more dimensions I think this way very quickly becomes much more difficult to visualize.