

Fibonacci
Phyllotaxis  Golden section/Golden Mean
 Golden Angle  Golden
angle vs. Fibonacci
Fibonacci Numbers  Golden Angle
Many plants display Fibonacci Phyllotaxis,
featuring Fibonacci numbers and the Golden Angle. The Golden
Angle is related to the Golden Mean,
itself a limit of quotients of Fibonacci
numbers. We review these facts below. The dynamical
model offers an explanation of why Fibonacci phyllotaxis is
so predominant. To find out more about the Fibonacci numbers and
the Golden Mean, visit Ron
Knott's web site.
Fibonacci Phyllotaxis 

In spiral
phyllotaxis, the number of visible spirals, called parastichies,
are most often two successive elements of the Fibonacci
sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...
where each number is the sum of the previous two. When this occurs,
the angle between successive leaves or botanical element is close
to the Golden Angle  about 137.5^{o}, related to the golden
mean.
The Aonium on the right exhibits 3 spirals winding
in one direction, 2 in the other. The angle between leaves 2 and
3 and the angle between leaves 5 and 6 are both very close to 137.5^{o}. 




Based on a survey of the literature encompassing 650 species and
12500 specimens, R. Jean (1994)
estimated that, among plants displaying spiral
or multijugate phyllotaxis about 92% of them have Fibonacci
phyllotaxis. 


Golden Section and Golden
Mean 
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The Greeks knew that to draw the pentagram,
one has to divide a segment in the Golden Section. The Golden
Section is the only way to divide a segment so that
the ratio of the large segment (Red here) over the small (Gold)
is the same as the ratio of the whole segment (Whole) over
the large (Red). This ratio is the Golden Mean
Ø (phi): 





Golden Section and Golden
Angle 
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One obtains the Golden Angle, of about 137.51 degrees,
by a golden section of the circumference of the circle:






Fibonacci Numbers and Golden
Mean 


The Fibonacci sequence 1, 1, 2,
3, 5, 8, 13, 21, 34 etc. is given by the recurrent relation
F_{n+1}= F_{n} + F_{n1 } (each term is
the sum of the two previous ones), with initial condition F_{0}=
F_{1}=1.
The Golden Mean is a limit of quotients of successive Fibonacci
numbers:
technically: Ø = lim
F_{n}/F_{n1
}as n goes to infinity (proof below) 
In the same way, the Golden Angle is given by the
limit of quotients of Fibonacci numbers that are two appart:
Golden Angle = 360(2 Ø)
= 360 lim F_{n}/F_{n+2
}
(proof below) So it is not surprising that the Fibonacci
numbers and the Golden Angle arise at the same time in plants. But
this is not an explanation of why and how they
do arise. The dynamical model does
provide an explanation, given general assumptions about the pattern
formation mechanism.



Return
to Top 
 To get the value
of Ø
from the golden section definition given
above, set Whole=x and Red=1. Then Gold=x1 and the equality Ø= Whole/Red =
Red/Gold becomes Ø= x
= 1/(x1), from which we obtain x(x1)=1, or x^{2}x1=0.
Using the quadratic formula, one finds that Ø
= x = (1+sqrt(5))/2 is the only positive
root of this equation.
 To prove that the Golden mean
is a limit of quotients of Fibonacci numbers, assume that lim
F_{n}/F_{n1}= L . We will show L= Ø.
Divide the equality F_{n+1}=
F_{n} + F_{n1} by F_{n
}to get F_{n+1}/F_{n} = 1+F_{n1}/F_{n}.
Taking the limit as n goes to infinity yields L = 1+1/L or L^{2}L1=0,
which has the only positive root L= Ø. A more rigorous proof
would not assume that L exists, as we did, but would start with
proving the equality (1Ø)^{n}= F_{n+1}F_{n}*Ø
by induction.
 To show that (2 Ø) =
lim F_{n}/F_{n+2
}, divide F_{n+1}=
F_{n} + F_{n1} by F_{n+1}
and proceed as before.
Fibonacci Phyllotaxis 
Golden section/Golden Mean  Golden
Angle  Golden angle vs. Fibonacci 

