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Why in nature, do most flowers have a Fibonacci number of petals?
Answer 1:

At the moment I don't agree with your statement - I think a lot of flowers have 4 petals or 6 petals. All the cruciferous plants, like mustard, have 4 petals. And I have seen a bush with white flowers that have either 5 or 6 petals, but I don't know its name. Eucalyptus pods have a pattern of 4- 5- or 6-pointed 'stars' on their flattish surfaces.

Answer 2:

For along time, it had been noticed that these numbers were important in nature and we still do not completely understand why. You also have to be aware that in the case of the flower petals the Fibonacci numbers are only verified on average while in the case of the sunflower, pineapple or pinecone the numbers are exact.

To understand why the number of petals correspond on average to a Fibonacci number we first have to look at a sunflower head. When you look at the head you notice two series of curves one winding in one sense and one in another; the number of spirals not being the same in each sense. Why is the number of spirals in general either 21 and 34, either 34 and 55, either 55 and 89, or 89 and 144? They all belong to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, but the explanation is also linked to another famous number, the golden mean. In many cases, the head of a flower is made up of small seeds which are produced at the center, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceding one. For example, if the angle is 90 degrees, that is 1/4 of a turn, the result after several generations is four straight lines starting from the center in a 90 degree angle.

Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc. (that is a simple rational number), one always obtains a series of straight lines. If you want to get a spiral, it is necessary to choose a portion of the circle which is an irrational number (or a non simple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly.

In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (see sunflower head).

When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc. These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower.

This is why the number of spirals in the centers of sunflowers, and in the centers of flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number.

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