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

1
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.

2
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.

3
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/

4
Figure 1: The branches of the tree and the Fibonacci sequence.

5
Figure 2: The reproduction of rabbits and the Fibonacci sequence
Figure 3: Number of spirals in a Sun
ower

6
Figure 4: In this gure is shown the spiral in another Sun
ower