Copie de Seed Heads and Spirals
This applet is for exploring the relationship between seed heads on flowers and spirals. You can animate it to watch the flower grow. The angle (measured in revolutions) between seed releases can be manipulated. Angles that are related to the Golden Ratio create nicely spread out seed heads, while angles that are rational with "small" denominators have lots of crowding and unused space.
The spirals that you see are really optical illusions. Two seeds that appear close to each other in a spiral were actually released far apart from each other with respect to time. You can manipulate if and how (up to two families of) these spiral curves are explicitly plotted. Detailed instructions are in the applet.
(All angles are measured in revolutions, not degrees or radians.)
What happens when you set the release angle to a "very rational" number like 2/5?
What is the continued fraction for 2/5? What are the partial convergents? How many "spirals" do you see amongst the seeds?
How many spirals do you see if you set the release angle to pi-3 = 0.1415926...? How does that relate to the partial convergents for this number?
What is the continued fraction for the golden ratio, and what are the partial convergents? Do you recognize their denominators? How do they appear in the seed head when the angle is the golden ratio?