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Phyllotaxis by TaffGoch Phyllotaxis by TaffGoch
Phyllotaxis spirals on sphere • Also described, in literature, as Fibonacci spirals • Natural spiral progression observed in the growth of leaves, pine cones, cacti, sunflower seed-heads, etc. • [Greek phyllo, leaf + taxis, arrangement]

2400 nodules • Nodule layout process, in Excel spreadsheet, employs the use of the "golden angle," to calculate, and subsequently import, cartesian x,y,z coordinates into Google SketchUp

Fibonacci Numbers - Golden Angle

Also, see: How I Made Wine Glasses from Sunflowers
(Before "wine glass" discussion, phyllotaxis spherical-distribution is described.)

3D-modeled in SketchUp • Rendered in Kerkythea

Best-quality PDF of Saff's and Kuijlaars' article, "Distributing Many Points on a Sphere," is (currently) posted at Arno Kuijlaars' personal webpages:  Saff-Kuijlaars-MathIntel97.pdf
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bugman123 Featured By Owner May 24, 2011  Hobbyist Digital Artist
Hey, that's great! It looks like you figured it out perfectly.
TaffGoch Featured By Owner May 24, 2011
Took me several months to find an appropriate online reference. Plenty of 2D methods available, but insufficient 3D methodology descriptions.

Found only ONE article, that mentions a relatively simple method, in the closing paragraphs of the article: "Distributing Many Points on a Sphere"
Nanaki-Murasaki Featured By Owner May 24, 2011  Professional Digital Artist
seems a tad silly.

if you have the Cartesian coordinates then adding the 3rd perspective of Z you should have been able to calculate this easily.
TaffGoch Featured By Owner May 24, 2011
Ah-ha, I thought the same thing, but doing that "bunches" the nodes at the poles, and "stretches" the node spacing, as you approach the equator.

I tried several times to get from 2D coordinates to 3D coordinates, with grossly unsatisfactory results.

The argument/solution only appears silly, until you actually try it. Your (and my initial) postulated mental-logic-solution doesn't prove out.

I wasn't sure the article method would work, until I tested it; seeing it validated and realized in 3D, using Excel/SketchUp.

"The proof of the pudding is in the eating."

lyc Featured By Owner May 24, 2011
thanks for the link, this is actually very useful for my work :D
TaffGoch Featured By Owner May 24, 2011
That is my hope -- that it will potentially inspire others, as it did me. :handshake:

Your servant,
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