Rotations of finite order are dense in SO(3)

Idea. An element of $SO(3)$ is determined by an axis and an angle. The axis is fixed during the approximation, and rational multiples of $2\pi$ are dense in the set of angles.

Step 1: Euler's theorem (every $U\in SO(3)$ has a fixed axis). The characteristic polynomial of $U$ is real of degree $3$. Its roots have modulus $1$ (because $U$ preserves the Euclidean norm, and $|Uv|=|\lambda||v|$ for an eigenvector $v\in\mathbb{C}^3$ with respect to the Hermitian norm, which $U$ also preserves). Non-real roots come in conjugate pairs $e^{\pm i\theta}$, so either all three roots are real, equal to $\pm1$, or there is one real root $\lambda=\pm1$ and a pair $e^{\pm i\theta}$. The product of the roots is $\det U=1$.
- All real: the product $1$ forces $(1,1,1)$ or $(1,-1,-1)$ up to order; in both cases $1$ is a root.
- One real root $\lambda$ and a pair: $\lambda\cdot e^{i\theta}e^{-i\theta}=\lambda=1$.

So $1$ is always an eigenvalue: there is a unit vector $n\in\mathbb{R}^3$ with $Un=n$.

Step 2: normal form. $U$ preserves $n^\perp$ (if $v\perp n$ then $\langle Uv,n\rangle=\langle Uv,Un\rangle=\langle v,n\rangle=0$). Choose an orthonormal basis $(u,v)$ of $n^\perp$ such that $(u,v,n)$ is positively oriented, and let $Q=[u\ v\ n]\in SO(3)$. In this basis $U$ is $\begin{pmatrix}V&0\\0&1\end{pmatrix}$ with $V\in O(2)$ and $\det V=\det U=1$, so $V=R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$. That is,
$$U=Q\begin{pmatrix}R(\theta)&0\\0&1\end{pmatrix}Q^T.$$

Step 3: approximation. Let $\theta_m=2\pi p_m/q_m$ with $p_m/q_m\in\mathbb{Q}$ and $\theta_m\to\theta$ (possible since $\mathbb{Q}$ is dense in $\mathbb{R}$). Define
$$U_m=Q\begin{pmatrix}R(\theta_m)&0\\0&1\end{pmatrix}Q^T\in SO(3).$$
It is the rotation about the same axis by an angle in $2\pi\mathbb{Q}$, and $U_m^{q_m}=Q\,\mathrm{diag}(R(2\pi p_m),1)\,Q^T=I$.

Because $Q$ is orthogonal, the operator norm is invariant under conjugation by $Q$ and
$$\|U_m-U\|=\|R(\theta_m)-R(\theta)\|=2\Bigl|\sin\frac{\theta_m-\theta}2\Bigr|\le|\theta_m-\theta|\to0.$$
(Indeed, $R(\alpha)-R(\beta)$ is the matrix of multiplication by the complex number $e^{i\alpha}-e^{i\beta}$ on $\mathbb{C}\cong\mathbb{R}^2$, whose operator norm is $|e^{i\alpha}-e^{i\beta}|=2|\sin\frac{\alpha-\beta}2|$.)

Check (numerical). For a random $U\in SO(3)$, rounding the angle to a multiple of $2\pi/N$ gives an error below $2\pi/N$ in operator norm, as predicted.

Remark. The set of finite-order rotations is uncountable (different axes) and each has finite order, but the *group* of all rotations by rational angles about one fixed axis is countable. This is typical of density arguments: a dense set is topologically large but not necessarily "large" in cardinality.

Lesson. Reduce to a normal form with continuous parameters (here the angle), then approximate the parameters inside the normal form.