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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    $\begingroup$ 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. $\endgroup$ Mar 9, 2013 at 5:51
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    $\begingroup$ Mandatory reference: youtube.com/watch?v=lOIP_Z_-0Hs :-) $\endgroup$
    – Mr.Wizard
    Mar 9, 2013 at 6:25
  • $\begingroup$ @Mr.Wizard her videos are awesome! $\endgroup$
    – chris
    Mar 9, 2013 at 14:49
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    $\begingroup$ here is the original picture if someone wants to give it a try: upload.wikimedia.org/wikipedia/commons/8/86/… $\endgroup$
    – shrx
    Dec 31, 2013 at 12:45

1 Answer 1


It is called a spiral phyllotaxis pattern. There is a lot of Mathematica material for this. Chris Carlson's blog entry gives you some 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}]]]

phyllotaxis pattern

The Demonstrations project has a lot too; for example:

Phyllotaxis Spirals, Stephen Wolfram

Phyllotaxis Explained, Claude Fabre

  • $\begingroup$ 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). $\endgroup$
    – Tony Dong
    Mar 9, 2013 at 23:42
  • $\begingroup$ @TonyDong I think you just need to count how many dots you place for full single angle turn 2Pi. $\endgroup$ Mar 10, 2013 at 0:39

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