Recently I've found a few different sources that all mention the mathematical pattern of spiral phyllotaxis, the spiral arrangement of leaves and petals of various plants.
The last paper is most relevant here, as it talks not just about how the spirals relate to the Fibonacci sequence, but also how the pattern develops as the inflorescence grows. It also links to animations of the growth pattern.
After reading that, I first explored the math of these spirals in Desmos and made this interactive graph.
This sketch is my artistic interpretation of the above. I create florets in the center of the circle, and each moves radially outwards. Each successive floret appears one golden angle away from the previous one. This is enough to produce the spiral pattern.
A real sunflower stops growing at a certain point, while my animation goes on forever. Also, a real flower would distinguish between petals and florets that produce seeds.