How sunflowers teach math

Eons ago, plants worked out the secret of arranging equal-size seeds in an ever-expanding pattern around a central point so that regardless of the size of the arrangement, the seeds pack evenly. The sunflower is a well-known example of such a “spiral phyllotaxis” pattern:

It’s really magical that this works at all, since the spatial relationship of each seed to its neighbors is unique, changing constantly as the pattern expands outwardly—unlike, say, the cells in a honeycomb, which are all equivalent. I wondered if the same magic could be applied to surfaces that are not flat, like spheres, toruses, or wine glasses. It’s an interesting question from an aesthetic point of view, but also a practical one: the answer has applications in space exploration and modern architecture.

To reproduce the flat sunflower pattern mathematically, you need to know three secrets of the arrangement:

1. Seeds spiral outward from the center, each positioned at a fixed angle relative to its predecessor.
2. The fixed angle is the golden angle, γ = 2π(1 – 1/Φ), where Φ is the golden ratio.
3. The ith seed in the pattern is placed at a distance from the center proportional to the square root of i.                           

The following photos simulate the spiral phyllotaxis pattern on mathematical objects..