Apologies for the vagueness of the question, I'll clean it up once an answer helps me describe it better.

I'm fascinated by the pattern demonstrated in this image. It's made up of dots on a series of concentric circles. The angles used, number of dots on each circle and circle sizes cause a spiral pattern to emerge.

Is there a name for this or a combination of principles at play here? I'm interested in the mathematics of it, and how such an image might be defined in equations.

Image credit: the talented fellow at http://dotboydesigns.vpweb.com.au

  • $\begingroup$ There was just a post here, invoking the golden ratio, with very interesting links, but when I returned to the page, the post was gone? I have pretty much all of the details; I'd be happy to re-post the post, as best I can, as a Community Wiki post. $\endgroup$ – Namaste Jun 17 '11 at 1:39
  • $\begingroup$ @amWhy: I had posted that answer, but I deleted it because upon closer inspection I felt that the image in the question did not actually use the golden ratio. I think I'll undelete it as you found it interesting. $\endgroup$ – Rahul Jun 17 '11 at 1:44
  • $\begingroup$ @Rahul: yes, please do repost! I found it fascinating! (I posted the remnants I had of your post (Community Wiki), then saw your comment, so deleted it...All yours! $\endgroup$ – Namaste Jun 17 '11 at 1:45
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    $\begingroup$ Not quite the same thing (or is it?), but equally puzzling: This is Not a Spiral and Spiral illusion. $\endgroup$ – lhf Jun 17 '11 at 1:50
  • $\begingroup$ It looks to me like it's dots taken straight from a spiral (rotated and copied multiple times), so it's not like the spirals are an emerging pattern, they're specifically constructed for in the graphic. That said, you can get into a mathematical argument over why the same spirals are seen both left-handed and right-handed. On the other hand, certain shapes in nature are logarithmic spirals, and this can be explained by Fibonacci numbers and the fact that the golden ratio describes when the Euclidean algorithm is least efficient: $(F_n,F_{n+1})$ ,$F_{n+1}/F_n \approx \phi$. Can't recall source. $\endgroup$ – anon Jun 17 '11 at 3:00

(I no longer think this is accurate; see the second paragraph.) The consecutive dots are rotated by the golden angle, which is $(1 - 1/\phi)$th of a whole turn, where $\phi = (1 + \sqrt 5)/2$ is the golden ratio. There's a nice interactive demonstration, as well as an explanation of how this works to create beautiful spirals, on this page: http://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html ...The essential reason is that the golden ratio is very poorly approximated by rational numbers, so no single spiral arises that can dominate the whole pattern.

Edit: I don't think the image in the question actually uses the golden angle. It's just lots of alternating dots in concentric circles, with some artistically chosen spreading of their radii. However, you can generate some very beautiful and very similar-looking spirals, without requiring any artistic selection, using the principles I described above.


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