# Understanding fibonacci ratio in plants

Recently, a 13year old kid has re-dicovered that there is a magic ratio for branching in plants. Following article describes his work in his own words.

http://www.amnh.org/nationalcenter/youngnaturalistawards/2011/aidan.html

In 1754, a naturalist named Charles Bonnet observed that plants sprout branches and leaves in a pattern, called phyllotaxis. Bonnet saw that tree branches and leaves had a mathematical spiral pattern that could be shown as a fraction. The amazing thing is that the mathematical fractions were the same numbers as the Fibonacci sequence! On the oak tree, the Fibonacci fraction is 2/5, which means that the spiral takes five branches to spiral two times around the trunk to complete one pattern. Other trees with the Fibonacci leaf arrangement are the elm tree (1/2); the beech (1/3); the willow (3/8) and the almond tree (5/13) (Livio, Adler).

I'm facing difficulty in understanding highlighted statement. He says fibonacci fraction for 2/5 means that spiral takes five branches to siral two times around the trunk.

Implies, Angle between each branch is 720/5 = 144 degrees.

But the kid says,

The pattern was about 137 degrees and the Fibonacci sequence was 2/5.

What am I not understanding?

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The article implies that, asymptotically, the spiral takes $F_n$ (the $n$-th Fibonacci number) branches to go $F_{n+2}$ times around the trunk of the tree. The fraction $2/5$ is only a first approximation to this phenomenon (the other cited approximations being $1/2,1/3,3/8,5/13$). The limiting ratio is $1/\phi^2$, where $\phi$ is the golden ratio $(1+\sqrt{5})/2$. Hence the rotation angle between branches should be approximately $360^\circ / \phi^2\approx137.507764^\circ$.

To see why the ratio is what it is, first let's define the ratio of successive Fibonacci numbers, in the limit, to be $$\phi = \lim_{n\to\infty}\frac{F_{n+1}}{F_n}.$$

(We'll leave aside the question of why this limit exists, i.e. why the Fibonacci numbers exhibit exponential growth.) Then, by the recurrence definition of Fibonacci numbers, we have

$$\frac{F_{n+1}}{F_n}=\frac{F_n+F_{n-1}}{F_n}=1+\left(\frac{F_n}{F_{n-1}}\right)^{-1}\to1+\phi^{-1}.$$

Hence $\phi$ solves the equation $x=1+1/x$. This can be multiplied through for the quadratic equation $$x^2-x-1=0.$$

The golden ratio $\phi$ must be the positive solution, $(1+\sqrt{5})/2$. Now note that $$\frac{F_n}{F_{n+2}}=\left(\frac{F_{n+2}}{F_{n+1}}\right)^{-1}\cdot\left(\frac{F_{n+1}}{F_n}\right)^{-1}\to\phi^{-1}\cdot\phi^{-1}=1/\phi^2.$$

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Thanks for explaining! –  claws Aug 25 '11 at 7:58