1Assignment u5s4The Fibonacci succession is the succession of numbers that, starting withunity, 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 treesThe Fibonacci sequence appears recurrently in nature, for example in howthe branches of the trees grow. If we look at a tree closely we see that thenumber of branches of the tree follows the Fibonacci sequence.

Consider this situation: Let’s suppose that we have a tree that grows therst year without making any new branch, but it generates a new branchat the second year and each year another branch. Each new branch followsthe same law. In the rst year the tree has one branch, the second yearthis branch has a new branch, therefore, in the second year the tree has twobranches.

In the third year the oldest branch has a new branch while theyoungest branch do not have any, therefore, in the third year the tree has3 branches. In the fourth year the oldest branch has a new branch and theyoungest branch has a new one (this branch is two years old), therefore, inthe fourth year the tree has 5 branches and so on. This process is shownin the gure one.

Note that the Fibonacci series allows us to predict thenumber of branches that the tree would have in a given time and vice versa.2Fibonacci and the RabbitsAnother interesting example is the following problem: how many rabbitswill there be after a certain number of months? To answer this question we consider the following situation: a farmerbuys a pair of rabbits (male and female). The couple of rabbits begin tohave bunnies at the second month of life and from that moment on thatcouple begin to procreate each month. Let’s suppose that the rabbits do notdie 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 rabbitsbut the youngest pair of not yet, therefore, in the third month we have threepair of rabbits. In the fourth month the rst pair of rabbit has other pair of rabbits, thesecond pair (that already are two months old) has another pair of rabbitsand 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 seriesof Fibonacci.Fibonacci and the SunowerThe sunower is an inorescence, it has hundreds or thousands of smallindividual owers with ve petals called orets. This orets are arrangedin the form of spirals that rotate clockwise and counterclockwise. Normally,there are 34 spirals in one direction and 55 towards the other; however, in avery large sunower head there could be 89 in one direction and 144 in theother, this pattern produces the most ecient seed packing possible withinthe ower head. Note that the number of spirals are terms of a Fibonaccisequence. In the gure 3 is shown the number of spirals in one direction andtowards the other in a Sunower and in the gure 4 we can apreciate thespiral in a real Sunower.

3Sources1. 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.

html2. https://simple.wikipedia.org/wiki/Sunower3. McGraw Victoria (2016).

7 Beatifull Examples of the Fibonacci Se- quence in Nature. https://www.theodysseyonline.com/7-beautiful-examples-bonacci-sequence-nature4. 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/4Figure 1: The branches of the tree and the Fibonacci sequence.5Figure 2: The reproduction of rabbits and the Fibonacci sequenceFigure 3: Number of spirals in a Sunower6Figure 4: In this gure is shown the spiral in another Sunower