Context: basecase.org/env/on-rainbows .
This animation helped me visualize how, by moving around a circle in strides of the golden angle, you partition it as smoothly as possible – for a fixed stride – as time increases indefinitely.
I think it’s equivalent to say that the golden angle is the only fixed stride such that every new division turns the largest gap into the smallest gap (and another gap). That’s just intuition; I haven’t proven it. [2011-12-22: See update at bottom!]
This has applications to plant leaf patterns, lug nut tightening order, color sequence generation, drum head tensioning, dart board numbering, and musical harmony, among other things – any situation where you want to distribute an ever-increasing number of things (like lug nut turns) around a circle. And it’s O(1) in time and space.
I cut it off at about 4200 steps, when no new pixels were getting drawn because I was too lazy to use a real graphics library. The code draws lines as series of pixels 1 Euclidian pixel apart, which is not continuous around the diagonals; thus the ugly moiré/aliasing interference patterns.
The original was 60 fps; Vimeo re-encoded it so you end up seeing more than one line appear per frame. If you eyeball the 137.5…° angle and have lots of time, you can figure out the order they were drawn in.
The title is in trochaic pentameter: Hów thē góldēn ánglē fílls ā círclē.
Update: Hacker News user cobbal prompted a nice explanation of this by Qiaochu Yuan on the math Stack Exchange: math.stackexchange.com/questions/93623/does-the-golden-angle-produce-maximally-distant-divisions-of-a-circle . The web is tuple space for ill-formed math ideas.