The golden flower

The opening drawing is a sum of turning circles:

$$z(t) = \sum_k r_k \, e^{i n_k t}, \qquad 0 \le t < 2\pi .$$

Each circle turns at its own whole-number frequency $n_k$ and carries the next on its rim; the pen at the end of the chain traces the curve. When every frequency leaves the same remainder 1 on division by $p$, turning the drawing through $2\pi/p$ maps it onto itself, so the flower has $p$ petals. The seven flowers in the rotation use three to five circles; the first has frequencies $1, -11, 13, -23$ and twelve petals.

Ptolemy's epicycles and a truncated Fourier series are the same construction. Resting the cursor on the drawing slows time; with reduced motion the finished flower is shown still.