Constructing some phyllotaxic surfaces from Sunflower 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. If so, We wondered the same magic could be applied to surfaces that are not flat, like spheres, toruses,….. 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.
Here is three secrets of the arrangement to reproduce the flat sunflower pattern mathematically:
- Seeds spiral outward from the center, each positioned at a fixed angle relative to its predecessor.
- The fixed angle is the golden angle, γ = 2π(1 – 1/Φ), where Φ is the golden ratio.
- The ith seed in the pattern is placed at a distance from the center proportional to the square root of i.
See more about technical analysis : How I Made Phyllotaxic Surfaces & Wine Glasses from Sunflowers by the author Christopher Carlson on http://blog.wolfram.com