fibonacci-numbers

. Shifting the register down the Fibonacci number results into the conversion from kilometers
to miles (Hazewinkel, 2002) (Knott, Using the Fibonacci numbers to represent whole numbers,
2009).
A close observation of our nature revels that Fibonacci numbers are often found in two
consecutive Fibonacci numbers in biological setting (Douady & Couder, 1995), branching in
trees, and arrangement of leaves on a stem, and the fruitlets of a pineapple (Jones & Wilson,
2006) for example. The flowering of artichoke and arrangements of a pine cone is yet another
example (Brousseau, 1969). The Fibonacci numbers have also been observed in the family tree
of honeybees (Thimbleby). H. Vogel in
1979 proposed a model for the pattern of
florets found in the head of a sunflower
(Vogel, 1979) as below:
2 , √
∅ Here, is the index number of the floret
while is a constant scaling factor.
Figure 3 Sunflower head displaying florets in spirals of 34 and 55
around the outside
Therefore, florets lie on Fermat's spiral. The divergence angle is approximately 137.51 ° which is
known as the golden angle because this angle divides a circle in the golden ratio. The reason
sunflower florets pack so efficiently is since this ratio is an irrational number; no floret has a
neighbor at exactly the same angle from the center. The rational approximations to the golden
ratio are of form : 1. For this reason, the nearest possible neighbor of floret number
are those at for some index which is dependent of , the distance from the center.
Fibonacci numbers Page 8