# 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.

The key take-away here is that unlike translation, where each new dimension simply adds one more degree of freedom of motion, each new dimension of space adds more opportunities for different kinds of rotation than the previous dimension did.