Each seed is placed a little further out and turned by a fixed fraction of a revolution from the last: seed $n$ sits at radius $r = c\sqrt{n}$ and angle $\theta = 2\pi n / R$. Choosing $R$ to be the golden ratio packs the seeds most evenly — the arrangement real sunflowers settle on.
The striking part is what the spirals mean. Whenever a fraction $p/q$ is close to $R$, seeds that are $p$ apart almost line up (after $p$ steps you have turned $p/R \approx q$ whole times), so you see $p$ spiral arms. The arm counts are therefore the numerators $p$ of the best rational approximations $p/q$ of $R$: the Fibonacci numbers for the golden ratio, and $22, 333, 355$ (from $22/7,\ 333/106,\ 355/113$) for $\pi$. A whole fraction like $13/8$ is exact, so it gives a single family — $13$ straight rays. Toggle the labelled spirals under the canvas to see them.