Rotations in three dimensions

Lie groups are continuous groups. An example of a continuous group, that I manage to visualize, are rotations in three-dimensional space.

Fig. 1: An arbitrary rotation is undone by three rotations about the z-, y- and x-axes. 

Rotations form a group since:

• The result of two rotations $a$ and $b$, conducted one after the other, written formally as $a \circ b$, is again a rotation ("closure").
• A series of rotations can be grouped at will, we'll always arrive at the same result. I.e. $(a \circ b) \circ c = a \circ (b \circ c)$ ("associativity").
• Rotating an object by zero degrees is regarded a rotation as well. This operation, denoted by $e$, commutes with every other rotation $e \circ a = a \circ e$ ("identity element").
• Every rotation through an angle $\alpha$ can be undone by a rotation through $-\alpha$ ("inverse element").

Every rotation in three-dimensional space can be decomposed into a rotation about the x-axis, a rotation about the y-axis and a rotation about the z-axis (Fig. 1).

Fig.2: Three-dimensional rotations are not commutative.

Rotations in three-dimensional space are non-commutative. I.e. in general the order of the rotations does matter for the final position (Fig. 2).

Algebraically rotations are expressed in terms of 3x3 matrices.
$$R_x(\alpha) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & -\sin\alpha & \cos\alpha \end{pmatrix} \\ R_y(\alpha) = \begin{pmatrix} \cos\alpha  & 0 & \sin\alpha \\ 0 & 1 & 0 \\ -\sin\alpha & 0 & \cos\alpha \end{pmatrix} \\ R_z(\alpha) = \begin{pmatrix} \cos\alpha  & \sin\alpha & 0 \\ -\sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
Here, the matrix $R_x(\alpha)$ describes the rotation about the x-axis by an angle $\alpha$. E.g., the point (x,y,z) rotated by $\alpha$ about the x-axis, by $\beta$ about the y-axis and by $\gamma$ about the z-axis is given by
$$R_z(\gamma) \cdot R_y(\beta) \cdot R_x(\alpha) \cdot \begin{pmatrix} x\\ y\\ z \end{pmatrix}$$