The Fibonacci numbers

A simple model for the population growth of rabbits is based on the following rules.

(a) The rabbits are always in pairs for reproduction.
(b) A rabbit pair must grow one generation to maturity and then begets one baby pair at every generation thereafter.
(c) No rabbit pairs die off.

Figure 8. Rabbit population growth by the Fibonacci numbers

Under these rules, we follow in figure 8 the rabbit population beginning from a baby pair of the 1^st generation. Since it takes one generation to maturity, there is an adult pair for the 2^nd generation, which is ready for reproduction. So, there are two rabbit pairs, the parent and baby pairs, of the 3^rd generation. Next, the adult pair begets a baby pair but the baby pair simply matures, so a family of three rabbit pairs for the 4^th generation, and so on. The numbers of total rabbit pairs at each generation are known as the Fibonacci numbers, named after Leonardo Fibonacci, a twelfth-century Italian mathematician who advocated the adoption of Arabic numbers. Denoting by F_n the Fibonacci number of n^th generation, then the set of Fibonacci numbers {F_n} is

{F_n}= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …}.

As in figure 8, in particular, F₁ =1 for the 1^st generation, F₂=1 for the 2^nd generation, F₃ =2 for the 3^rd generation, F₄=3 for the 4^th generation, and so on. It is interesting to point out that the Fibonacci numbers F_n obey the following relationship

F_n = F_n-1 + F_n-2,

That is, F_n is given by sum of the two previous Fibonacci numbers, F_n-1 and F_n-2 For this relationship to work correctly, we require the index n be 3 or greater. For example, F_{3 =} F₂ + F₁ = 1+1 = 2, and we also see that F_{4 =} F₃ + F₂= 2+1 = 3  and F_{10 =} F₉ + F₈ = 34+21 =55. Lastly, we observe Fibonacci numbers in the petals of some common flowers. For instance, the lilies and irises have 3 petals, primroses and buttercups have 5 petals, corn marigolds have 13 petals, and daisies can have 34, 55 and 89 petals.

We show in Project d that the ratio of two adjacent Fibonacci numbers approaches the golden ratio after many generations; that is, F_n / F_n-1 as n becomes large. This is indeed a mystery. What does the golden ratio have to do with a rabbit population model in figure 8?

[prev] [ 0 | 1 | 2 | 3 | 4 | 5 | 6] [next]

back to the lesson list