Was Georg Simmel right in his mathematical argument against Nietzsche? As you can see in this link, the following argument originated from Georg Simmel was put forward against Nietzsche:


  Even if there were exceedingly few things in a finite space in an
  infinite time, they would not have to repeat in the same
  configurations. Suppose there were three wheels of equal size,
  rotating on the same axis, one point marked on the circumference of
  each wheel, and these three points lined up in one straight line. If
  the second wheel rotated twice as fast as the first, and if the speed
  of the third wheel was 1/π of the speed of the first, the initial
  line-up would never recur.


I'm not here to ask about the philosophy, but rather the validity of Simmel's argument. Can't they really recur back to their original state even if it takes so much time?

My take: Even though one of the wheel's speed is irrational, it's still a constant, so I think we can pass his assumption of having an irrational number as a wheel's speed. Having that, shouldn't we be able to reach a point where they will end up back to their original state like many least common multiple problems do (although as I have found while trying to solve it, it's not as those problems)?
If you just look at the first wheel going 1 rotation per unit time, and the other wheel going $1/\pi$ rotations, they will never again line up (I do not know why that example has 3 wheels).   

Let $r_1(t)$ and $r_2(t)$ be the radians spun by the first and other wheel over time $t\geq 0$.  Then: 


$r_1(t) = 2\pi t$
$r_2(t) = \frac{2\pi t}{\pi} = 2 t$


An integer number of rotations means an integer multiple of $2\pi$ radians. So to line up again at some time $T&gt;0$, we need positive integers $n,k$ so that

\begin{align}
\underbrace{r_1(T)}_{2\pi T}&amp;= 2 \pi k\\
\underbrace{r_2(T)}_{2T}&amp;= 2\pi n
\end{align}

So we need $2 \pi T = 2 \pi k, 2 T = 2\pi n$, so $T = k = \pi n$, that is: 
$$ \pi = \frac{k}{n} $$
This is impossible since $\pi$ is irrational.
Atleast in the example given above George  Simmel is wrong. Can u really give a wheel the speed of 1/pi? Since pi is an irrational eventually you'd have to truncate the (1/pi) number upto certain decimal point This will  hence make the speed rationally representable. And because of that the wheels eventually will coincide.
The main question is ,does there exist a physical process that can impart a non rational speed to an object.If answers is yes, then  (only for this example) Simmel would be correct