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I'm want to generate a 2D spiral pattern like the the one shown in the following figure. I can generate the spirals one by one, but I think there ought to be a more efficient method for generating such a scattered point pattern. Could anybody give me a hint?

sunflower pattern

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I think you need to show your existing approach to be in with a chance of getting an answer to this. As it is, it's rather unclear what you expect other than for people to solve the problem for you. –  Oleksandr R. Mar 9 '13 at 5:51
Mandatory reference: youtube.com/watch?v=lOIP_Z_-0Hs :-) –  Mr.Wizard Mar 9 '13 at 6:25
@Mr.Wizard her videos are awesome! –  chris Mar 9 '13 at 14:49
here is the original picture if someone wants to give it a try: upload.wikimedia.org/wikipedia/commons/8/86/… –  shrx Dec 31 '13 at 12:45
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1 Answer

up vote 19 down vote accepted

It is called spiral phyllotaxis pattern. There is a lot of Mathematica material for this. Chris Carlson blog give you a code right from the start:

g = 2 Pi (1 - 1/GoldenRatio);
PolarCoordinate[r_, t_] := r {Cos[t], Sin[t]};
Graphics[Point[Table[PolarCoordinate[Sqrt[i], i g], {i, 1, 1000}]]]

enter image description here

Demonstrations project has a lot too, for example:

enter image description here

enter image description here

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I have a question though: how to determine the number of spirals in such a pattern for a given phi (here g)? Claude Fabre showed that all points locate on a single spiral curve, but the pattern here seems several spirals exist as well as depicted in the sunflower pattern (cyan curves). –  Tony Dong Mar 9 '13 at 23:42
@TonyDong I think you just need to count how many dots you place for full single angle turn 2Pi. –  Vitaliy Kaurov Mar 10 '13 at 0:39
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