MODEL A SUNFLOWER WITH THE GOLDEN RATIO
It is well-documented that the Golden Ratio is observed in the angular geometry of seeds in sunflowers, coneflowers and pinecones, as well as other plants and natural phenomena. (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html)
However, little regard has been paid, it seems, to the distance of each seed from the center. When I looked at sunflowers and coneflowers, I noticed that the density of seeds is not uniform throughout. Seeds toward the center are smaller than the seeds near the edges, and thus, their density is greater.
This file makes a connection between the Golden Ratio and the seed distance from the center of the sunflower. This accounts for the growth of each seed over time, and provides a better model for seed location than linear distance from center.
AUTHOR: Joseph Kirk (c) 5/2006 EMAIL: jdkirk630 at gmail dot com
phi = (sqrt(5)-1)/2; % = 0.6180339887499
n = 2618;
The seed distances are raised to the Golden Ratio power
rho = (0:n-1).^phi;
A cirlce contains 2*pi radians, so 2*pi*phi cuts the circle by the Golden Ratio
theta = (0:n-1)*2*pi*phi;
polar(theta, rho, 'b.'); title([num2str(n) ' Sunflower Seeds']); set(gcf, 'color', 'w');