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I had to modify the question to get into a clearer answer. I originally asked this:

Original question

Watching some TV series, I saw the claim that all flowers have a number of petals that is related to the Fibonacci sequence (1,2,3,5,8,13,....). I was wondering if this is always true and if so, why? I guess some flowers have defects, this has to be some kind of probabilistic law.

I found some websites with examples, [1]. The usual explanation claims that is an optimal packing and is somehow related to petals forming a spiral.

Looking a bit deeper, I found that to avoid petals growing in the same positions, the petals can arrange in such a way that the angle between each petals is the circumference $2\pi$ divided by an irrational number $r$ in order to cover the circle densely. The golden mean $\phi=(1+\sqrt{5})/2$ is irrational and is related to Fibonacci sequence. However, even if $\phi$ is irrational, that does not explain why would $r$ be proportional to $\phi$ in order to create the spiral, wouldn't any irrational number work?

Could anybody provide additional understanding on why the golden mean is so important here and if possible what is the biological origin of these sequences?

More insight

Some comments below were arguing that there are some clear exceptions to this rule. One answer even showed a list of flowers categorized by different number of petals (1,2,3,4,5,6,7...) which shows some violations to the rule. Others provided additional sources that claim slightly different observations but I cannot see which are right and which aren't.

Most websites agree that daisies and sunflowers inner small petals (or disc flowers) produce Fibonacci spirals. Do the external petals (or ray flowers) also count? Other links add more on phyllotaxis (the number of arrangements of petals/leafs in a plant) and consider spirals in pinecones and pineapples. Maybe it is only related to spiral patterns in plants?

I am still concerned to understanding if there is any common mechanism and why the golden ratio is important here.

In this page of UC Santa Barbara, it says that the angle is about 137.5 degrees which is equivalent to the complement of $2\pi/\phi$ or equivalently to $2\pi/\phi^2$ due to the properties of $\phi$. In this paper introduction Phyllotaxis: is the golden angle optimal for light capture? it is claimed that two other angles work too (including the Luca angle). Note that that makes the question harder to falsify, as for example the Luca sequence also includes additional numbers like 4 and 7, but I guess the important thing is some kind of ratio and not the total number of petals/flowers in a given flower/plant.

So a better question is, when and how is phyllotaxis related to the Fibonacci sequence?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Bryan Krause
    Aug 25 at 17:24
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The claim that "all flowers have a number of petals that is related to the Fibonacci sequence" is simply false. Many flowers have other numbers of petals.

Consider, for example, dogwoods, which have a very clear four-petal form. Lots of other flowers have four or six. A nice catalog of some common examples can be found on this "Ontario wildflowers" site, which lists by petal count.

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    $\begingroup$ The claim is still somewhat true for a certain subset of flowers as discussed in the comments above. $\endgroup$
    – Mauricio
    Aug 23 at 13:28
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    $\begingroup$ @Mauricio I think it's on you to show that it's actually interesting, though, or drill down to a better description of "when it's true". Birthdays follow the Fibonacci, too, as long as you exclude everyone that doesn't fit the pattern. $\endgroup$
    – Bryan Krause
    Aug 23 at 13:30
  • $\begingroup$ @Mauricio - Are you saying there are no falsehoods on the internet? Dead. Horse. $\endgroup$ Aug 23 at 20:20
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    $\begingroup$ The claim that is false is: "all flowers have a number of petals that is related to the Fibonacci sequence." Given how much the question has changed, I have edited the answer to make it clear what I am addressing. $\endgroup$
    – jakebeal
    Aug 24 at 11:21
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    $\begingroup$ I think this argument is not really going to go anywhere. The general claim is obviously false. jakebeal's answer is simple and clear. It may be worth looking at the extent to which specific flowers or parts of flowers follow Fibonacci sequences, but if we're going to do that we need to start with a specific question with a narrow claim about what is supposedly following this pattern, not "flowers", or this turns into a numerology game rather than science. $\endgroup$
    – Bryan Krause
    Aug 25 at 17:01

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