MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
One natural analog is the answer to this previous question… – Gjergji Zaimi May 14 '12 at 13:24
@Gjergji: Yes, that's what got me thinking in this direction. But I thought there might be a more direct analog. Perhaps a vain hope... – Joseph O'Rourke May 14 '12 at 13:34
What would happen if one took a pair of irrationals, $(\alpha_1 ,\alpha_2)$ acted on a torus, and then took an orbit. Then pick a two meridians of the torus, collapse them to a point, then took the image of an orbit in the resulting 2-sphere? – Spice the Bird May 15 '12 at 3:35

If you take two random elements in $SO(3),$ (this is sort of like irrationals in $SO(2)$) the group they generate is both free and equidistributed in $SO(3),$ so the orbit of the north pole (or any other point you favor) will be equidistributed in $\mathbb{S}^2.$ You may complain that a free group will need two indices, but since free groups are orderable, you can make the two indices into one.

share|cite|improve this answer
Nice, Igor---Thanks! [And I would never so complain. :-)] – Joseph O'Rourke May 14 '12 at 15:15
This may reveal my ignorance, but is not the composition of two rotations itself a rotation, so that there will be nonuniform clustering near the axis of that rotation? – Joseph O'Rourke May 14 '12 at 17:31
The composition of two rotations is a rotation, but different words in the group will have different axes... – Igor Rivin May 14 '12 at 17:34

Kuipers and Niederreiter, Uniform Distribution of Sequences, give these references for questions of uniform distribution of sequences on a sphere:

V I Arnol'd, A L Krylov, Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex region, Dokl Akad Nauk SSSR 148 (1963) 9-12; English translation, Soviet Math Dokl 4 (1963) 1-5.

P Gerl, Gleichverteilung auf der Kugel, Arch der Math 24 (1973) 203-207.

Those of us whose familiarity with kugel extends only to the side dish commonly served on the Shabbes and other holidays ( may find some amusement in the title of Gerl's paper.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.