# 1 Assignment u5s4 The Fibonacci succession is the succession of numbers that

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Assignment u5s4

The Fibonacci succession is the succession of numbers that, starting with

unity, each of its terms is the sum of the previous two. The rst ten terms of the sequence of Fibonacci are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

Fibonacci and the trees

The Fibonacci sequence appears recurrently in nature, for example in how

the branches of the trees grow. If we look at a tree closely we see that the

number of branches of the tree follows the Fibonacci sequence.

Consider this situation: Let’s suppose that we have a tree that grows the

rst year without making any new branch, but it generates a new branch

at the second year and each year another branch. Each new branch follows

the same law. In the rst year the tree has one branch, the second year

this branch has a new branch, therefore, in the second year the tree has two

branches. In the third year the oldest branch has a new branch while the

youngest branch do not have any, therefore, in the third year the tree has

3 branches. In the fourth year the oldest branch has a new branch and the

youngest branch has a new one (this branch is two years old), therefore, in

the fourth year the tree has 5 branches and so on. This process is shown

in the gure one. Note that the Fibonacci series allows us to predict the

number of branches that the tree would have in a given time and vice versa.

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Fibonacci and the Rabbits

Another interesting example is the following problem: how many rabbits

will there be after a certain number of months? To answer this question we consider the following situation: a farmer

buys a pair of rabbits (male and female). The couple of rabbits begin to

have bunnies at the second month of life and from that moment on that

couple begin to procreate each month. Let’s suppose that the rabbits do not

die and that each female procreates a new pair of rabbits. In the rst month the couple of rabbits can not procreate yet, therefore,

we have two rabbits. In the second month the couple of rabbits has a pair of bunnies, therefore,

we have two pair of rabbits.

In the third month the oldest couple of rabbits has a new pair of rabbits

but the youngest pair of not yet, therefore, in the third month we have three

pair of rabbits. In the fourth month the rst pair of rabbit has other pair of rabbits, the

second pair (that already are two months old) has another pair of rabbits

and the third pair of the rabbits not yet (are only one month old), therefore,

in the fourth mounth we have 5 pair of rabbits and so on.. In the gure two we show this situation.

You can see that the number of pairs of rabbits per month form a series

of Fibonacci.

Fibonacci and the Sun

ower

The sun

ower is an in

orescence, it has hundreds or thousands of small

individual

owers with ve petals called

orets. This

orets are arranged

in the form of spirals that rotate clockwise and counterclockwise. Normally,

there are 34 spirals in one direction and 55 towards the other; however, in a

very large sun

ower head there could be 89 in one direction and 144 in the

other, this pattern produces the most ecient seed packing possible within

the

ower head. Note that the number of spirals are terms of a Fibonacci

sequence. In the gure 3 is shown the number of spirals in one direction and

towards the other in a Sun

ower and in the gure 4 we can apreciate the

spiral in a real Sun

ower.

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Sources

1. Dr Knott Ron (1996-2016). Fibonacci Number and Nature. Departa- ment of the university of Surrey, UK. http://www.maths.surrey.ac.uk/hosted-

sites/R.Knott/Fibonacci/bnat.html

2. https://simple.wikipedia.org/wiki/Sun

ower

3. McGraw Victoria (2016). 7 Beatifull Examples of the Fibonacci Se- quence in Nature. https://www.theodysseyonline.com/7-beautiful-examples-

bonacci-sequence-nature

4. Connor Thomas (2015). The Roundup. 1,1,2,3,..GO! Fibonacci Day at Jesuit https://www.jesuitroundup.org/news/academics/math/1123-go-

bonacci-day-at-jesuit/

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Figure 1: The branches of the tree and the Fibonacci sequence.

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Figure 2: The reproduction of rabbits and the Fibonacci sequence

Figure 3: Number of spirals in a Sun

ower

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Figure 4: In this gure is shown the spiral in another Sun

ower