Let $\operatorname{frac}(x) = x - \lfloor x \rfloor$ be the fractional part of $x$.
Then, for $\alpha$ irrational, $\operatorname{frac}(n \alpha)$, $n=1,2,\ldots$, distributes
randomly in $[0,1)$, which can be viewed as the circle $\mathbb{S}^1$.
Weyl's Equidistribution Theorem establishes the uniformity of the distribution.

_{$\operatorname{frac}(n \pi)$ for $n=1,2,\ldots,100$.}

Is there an analog to $\operatorname{frac}(n \alpha)$ for $\mathbb{S}^2$? Is there a function $f(n,x)$ with the behavior that, for some point $x \in \mathbb{S}^2$, $f(n,x)$ for $n=1,2,\ldots$ fills $\mathbb{S}^2$ randomly and uniformly?

A pointer would suffice if this is well known. Thanks!