The terms of the famous “Fibonacci sequence” are frequently found in nature, according to our partner The Conversation.
On pine cones, pineapples or sunflowers, we observe patterns in the form of spirals which are organized in two crossed networks.
And if we count the spirals of these networks, we very often get two consecutive numbers of the Fibonacci sequence.
The analysis of this phenomenon was carried out by Gaëlle Chagny, CNRS researcher in mathematics (statistics) and Thierry de la Rue, CNRS researcher in mathematics.
1, 1, 2, 3, 5, 8, 13, 21, 34…
The most discerning readers will have recognized the beginning of the famous Fibonacci sequence, in which each new term is obtained as the sum of the two previous terms, and known for its links with the mythical golden ratio.
Exact value of the golden ratio © G. Chagny & T. de la Rue via The Conversation
At the beginning of the 13th century, when Leonardo Fibonacci introduced this sequence in his treatise "Liber Abaci" to model in a very simplified way the evolution of a population of immortal rabbits, he had no idea of the importance it would acquire in mathematics, to the point that a scientific journal will be entirely devoted to it a few centuries later (
The Fibonacci Quarterly
, created in 1963).
He had probably not understood either that the terms of this sequence are frequently found in nature, not in the study of the demography of rabbits… but in botany!
On pine cones, pineapples, or flowers of the sunflower family, we observe patterns in the form of spirals, which are organized into two intersecting networks.
If curiosity leads us to count the spirals of these networks, we very often obtain two consecutive numbers of the Fibonacci sequence.
For example on a pine cone there are generally 8 spirals in one direction and 13 spirals in the other direction.
Pine cone © G. Chagny & T. de la Rue via The Conversation
>
At the heart of a daisy or an aster, the tiny flowers arranged on the capitulum (the florets) form two families of 13 and 21 spirals, or even 21 and 34.
On larger flowers like sunflowers, we find the pairs (34.55) or (55.89), and possibly more.
Aster © G. Chagny & T. de la Rue via The Conversation
Daisy variety © G. Chagny & T. de la Rue via The Conversation
It seems hard to believe in a simple coincidence: the regularity of this phenomenon seems rather to reveal the implementation of mathematical principles in the complex mechanisms of plant growth.
It was work on phyllotaxis, the science that studies the (geometric) arrangements of the leaves, flowers or petals of plants, which highlighted these principles from the 19th century.
Mathematicians and computer scientists then looked into the question, and worked on models to describe the mechanisms of plant formation in a simplified way.
A pattern for sunflower flowers
We have produced an online application that shows the operation of one of these models, based on the work of Helmut Vogel.
The same type of model is used in this excellent video (in English) from Numberphile, which we also drew inspiration from for the explanations that follow.
This model is based on a general principle: the organs of a plant such as the leaves, the petals, or here the florets, grow one after the other, each time forming a fixed angle with the previous one.
This angle is called the angle of divergence, and it is usually expressed as a proportion of a full turn.
Thus, an angle of divergence which is worth 1/6 means that one turns one-sixth of a turn around the center of the capitulum to place the next floret.
If the position of the new floret after this rotation comes to overlap with an already existing floret, the distance from the center of the flower head is increased so as to deviate sufficiently from the old floret.
Start of construction for a divergence angle of 1/6 © G. Chagny & T. de la Rue via The Conversation
In this model, the process ends when one arrives at the edge of a disc whose size is fixed in advance.
What figure do we get then?
Its shape depends strongly on the value of the divergence angle.
Let's first see the result for simple values.
Simulations for ¼ and 1/6 © G. Chagny & T. de la Rue via The Conversation
Simulation for 11/23 © G. Chagny & T. de la Rue via The Conversation
Branches in the flowers
When you turn a quarter turn (first image), 4 straight branches form, which is easily explained: if you repeat this operation 4 times, you come back in the starting direction. Similarly, a sixth of a turn gives 6 branches (second image). When the angle of divergence is equal to 11/23 (third image), 23 iterations of the rotation correspond to 11 complete turns, therefore also lead to the starting direction: we then obtain 23 rectilinear branches.
From the plant's point of view, these angles of divergence are not really interesting, because a good part of the space available on the flower head is not exploited.
This problem arises each time we choose a rational angle of divergence, that is to say one that is written as the quotient of two integers.
It is therefore tempting to test what happens when the angle of divergence is not rational.
We then say that it is irrational: we cannot write it as the quotient of two integers.
A first example of an irrational number that comes to mind is that of the famous number ϖ, which occurs in the calculation of the perimeter of the circle and in so many other mathematical formulas.
The following figure shows the result obtained if we choose a divergence angle equal to 1/ϖ.
Simulation for 1/ϖ © G. Chagny & T. de la Rue via The Conversation
The number ϖ is too rational!
Curiously, while ϖ (therefore also 1/ϖ) is not a rational number, the figure is of the same type as that obtained for a rational like 11/23, here with 22 branches.
Nevertheless these branches are here slightly curved.
Careful examination of the figure given for 1/ϖ will allow us to better understand the expected qualities of the angle of divergence for the plant.
Let's start by observing what happens towards the center: the process begins by forming 3 curved branches.
We can easily explain the presence of these 3 branches: the decimal writing of ϖ begins with 3.14159… so in a first approximation, ϖ is close to 3. The number 1/ϖ is therefore close to 1/3.
However, an angle of divergence 1/3 would give a figure with 3 rectilinear branches, and what happens towards the center is therefore an approximation of this behavior.
These 3 branches are curved here, because 1/ϖ is not exactly equal to 1/3.
Then, as new florets are formed, the distance to the center increases, and approaching the angle of divergence by 1/3 becomes too coarse.
The 3 central branches then end up disappearing.
How then to explain the formation of 22 new branches?
Another simple approximation of the number ϖ, much more precise than 3, is given by the fraction 22/7≃3.142857.
Thus the angle of divergence is very well approximated by the fraction 7/22.
If it were equal to 7/22, we would observe 22 straight branches.
As the approximation is very good but not exact, we obtain these 22 slightly curved branches.
Even if 1/ϖ is not a rational number, it is far from optimal, because too much space is lost on the flower head: on the same surface, we could put many more florets which would give more seeds!
The lost surface comes from the fact that the angle of divergence 1/ϖ is too well approximated by rational numbers.
For the florets to be placed very effectively on the flower head, it would therefore be necessary to use an angle of divergence which is as badly as possible approximated by rational numbers.
The golden ratio comes into play
The problem of Diophantine approximation (how to approximate any number by rational numbers) is related to the theory of continued fractions, which consists in representing any number in the form of a fraction, possibly infinite when the number is irrational.
For example, the number 1/ϖ can be written in the form where the fraction continues indefinitely.
Writing the inverse of ϖ as a continued fraction © G. Chagny & T. de la Rue via The Conversation
By truncating this infinite fraction, we obtain the best approximations of an irrational number given by rationals: what are called the reduced ones.
For example, by stopping the fraction at the number 7, we obtain the reduced 7/22 as a good approximation of 1/ϖ.
Approximation of the inverse of ϖ by one of its reduced © G. Chagny & T. de la Rue via The Conversation
However, there is a number for which these reductions remain as far apart as possible: the one whose continued fraction expansion only includes “1s”.
It is written
Continued fraction for the inverse of the golden ratio © G. Chagny & T. de la Rue via The Conversation
But this number is precisely the inverse of the golden ratio:
The inverse of the golden ratio © G. Chagny & T. de la Rue via The Conversation
This theory thus leads us to the optimum choice of the angle of divergence, the one which is the worst approached by rational numbers: it is the angle whose ratio with a complete revolution is the inverse of the golden ratio, and which is called the golden angle.
Let us see the figure given in this case.
Simulation when the angle of divergence equals the golden angle 1/φ © G. Chagny & T. de la Rue via The Conversation
We actually observe with the golden angle a much better use of the available space: the florets are arranged regularly without leaving a large empty space on the flower head.
There is also a real resemblance to the arrangement observed on real sunflowers.
And Fibonacci in all this?
The spirals in the figure of the golden angle correspond to the "branches" observed in cases of an (almost) rational angle of divergence: each network of spirals is associated with a rational approximation of the angle gold by one of its reduced.
As in the cases 1/6, 11/23 or 1/ϖ, the number of spirals (or branches) is given by the denominator of the fraction defined by the reduced.
However, the reductions of the golden angle have as successive denominators… the Fibonacci numbers!
Our "MATHEMATICS" file
Thus the presence of Fibonacci numbers in the spirals of plants is the signature of the fascinating golden number.
Its particular difficulty in being approached by fractions has led it to be selected by the natural evolution of many plants.
It is remarkable to note that mathematical theories a priori very far removed from botany shed relevant light in this field.
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This analysis was written by Gaëlle Chagny, CNRS researcher in mathematics (statistics) and Thierry de la Rue, CNRS researcher in mathematics, both at the University of Rouen Normandy.
The original article was published on The Conversation website.
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