When do polar roses $r = \sin k\theta$ start looping?

Question

I tried mapping on Desmos.com different roses $r = \sin k\theta$. As I vary $k$, I get all sorts of pretty roses, but I wonder at what value of $\theta$ they start "looping". For integer values of $k$, I know that for even $k$ I get $2k$ petals and they start looping at $k=2\pi$ and for odd I get $k$ petals and they start looping for $k=\pi$. But what if $k$ is not an integer? I tried $k=5.9$ and the flower stops changing around $37.7$. I noted that $5.9\cdot2\pi = 37.07$, which is close, but not quite there. For $k = 5.8$, the flower stops changing around $\theta=15.8$. Is there a way to find out exactly when the rose will stop changing for an arbitrary value of $k$?

Answer 1

This information is from Wikipedia. The rose will loop iff $k$ is rational. Now given a rational $k=\frac nd$ in irreducible terms:

• If $n$ and $d$ are both odd the rose loops at $\theta=d\pi$.
• If they have different parities the rose loops at $\theta=2d\pi$.

In your cases, with $k=5.9=\frac{59}{10}$ the rose loops at $\theta=20\pi$ – that you said it was closer to $10\pi$ is merely your lack of resolution of the very dense rose centre. For $k=5.8=\frac{29}5$ the rose loops at only $\theta=5\pi$.