# Is there any relation between phyllotaxis and the Fibonacci sequence?

## 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, . 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?

• Comments are not for extended discussion; this conversation has been moved to chat. Aug 25 at 17:24