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Suppose that you are given two rotations $R_1$ and $R_2$ in $SO(3)$ with the following properties:

$R_1$ is a rotation by an irrational multiple of $\pi$ about the z-axis.

$R_2$ is a rotation by a rational multiple of $\pi$ about an axis which lies at an angle $\phi$ from the z-axis, where $0<\phi<\frac{\pi}{2}$.

Do $R_1$ and $R_2$ densely generate $SO(3)$?

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Note that this is a follow up to a previous question:… – Adam Bouland Jan 27 '12 at 17:59
up vote 9 down vote accepted

Yes. Let $G$ be the closed subgroup of $SO(3)$ generated by those rotations. The complete list of closed subgroups of $SO(3)$ is known. It is

Finite groups: The cyclic group $\mathbb{Z}/n$, the dihedral group $D_{2n}$, the symmetries of the tetrahedron, the symmetries of the cube, and the symmetries of the dodecahedron.

One dimensional subgroups: $\mathbb{R}/\mathbb{Z}$, and $\mathbb{Z}/2 \ltimes \mathbb{R}/\mathbb{Z}$. The former is the group of all rotations about a fixed axis. The latter is the group of all rotations that take that axis to itself, meaning both rotations about that axis and rotations by $180^{\circ}$ about an axis perpendicular to it.

Three dimensional subgroups: All of $SO(3)$.

$G$ can't be finite because it contains an irrational rotation. It can't be one dimensional the angle between your rotations' axes is neither $0$ nor $\pi/2$. So it is all of $SO(3)$, as desired.

EDITED because I left a case off the list before.

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Thanks! Do you know a good reference for this characterization of subgroups of $SO(3)$? – Adam Bouland Jan 27 '12 at 22:19

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