The Fibonacci sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 12...
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The Fibonacci sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, ….
This famous sequence of numbers is present in a variety of fields: in art, in nature, botany, zoology, but especially in relation to the golden ratio phi and the golden spiral. Made its appearance in the "Liber Abaci", but centuries earlier had already been considered by the Indian mathematician Virahanka and described in 1133 by the scholar Gopal, as a solution to a problem of metrics related to poetry.
Fibonacci developed his suquence to solve the following problem concerning the breeding of rabbits:
"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from the initial torque in a year assuming that in a month each pair produces a new pair can reproduce itself in the second month? "
To solve this question Fibonacci assumed that each pair of rabbits:
a) starts to generate after the first month of age
b) generates a new pair every month
c) never dies.
He proceeded by considering a single pair that after the first month becomes mature and generates another couple. After the second month in a mature couple produces another young couple while the former becomes mature young couple (couples are then three). After the third month in each of the two mature couples generates a new request while the young couple becomes mature, so couples are five. After the fourth month the three mature couples each generate a new pair and the two young couples become mature. At this point, it is now clear how one can calculate the total number of pairs in each month but also the number of couples young and adult ones. In turn, the number of young couples to mature couples generate a Fibonacci sequence.
At this point analyzing the diagram below we can see how the numbers of pairs in each month go to form the Fibonacci sequence:
Pattern inherent to the problem of rabbits in orange are represented mature couples, young ones in blue. The definition of Fibonacci sequence
Taking the cue from the previous issue of the rabbits, and extending, the Fibonacci sequence can be defined as follows: the first two elements are 1, 1; every other element is the sum of the two preceding it.
Calling F (n) the Fibonacci sequence, we have the following mathematical definition: F(1) = 1 F(2) = 1 F(n) = F(n-2)+F(n-1) per n = 3, 4, 5, ...
According to this definition it is assumed conventionally F(0) = 0.
So the sequence of Fibonacci:
0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Note that the function F (n) is recursive, that is defined in terms of the function itself. The particularities of the sequence
The sequence is characterized by numerous and curious feature:
3.1 The square of a Fibonacci number less than the square of the second number is always a previous number of the sequence
3.2 The greatest common divisor of two Fibonacci numbers is still a Fibonacci number
3.4 Adding an odd number of products of successive numbers in the sequence, the three products as 1x1, 1x2, 2x3, you get the last square Fibonacci number present in the products in question. Indeed (1x1) + (1x2) + (2x3) = 2 + 1 + 6 = 9, is the square of the last number that appears in the previous product (in this case 3). Similarly, we can analyze the series of seven products: (1x1) + (1x2) + (2x3) + (3x5) + (5x8) + (8x13) + (13x21) = 1 + 2 + 6 + 15 + 40 + 273 +104 = 441 which is just the square of the last number that appears in the product. This property can be represented geometrically as shown by the figure:
An odd number of rectangles with sides equal to a number of terms of the Fibonacci sequence are exactly placed in a square the side of which coincides with a side of the larger rectangle.
3.5 The sequence is also connected with the triangle Tartaglia which is a geometric arrangement in the shape of a triangle of binomial coefficients, is the coefficients of the expansion of the binomial (a + b) raised to any power n.
From this triangle can be drawn Fibonacci numbers, adding the numbers of the diagonals as shown in the figure: so we get from the first line 1, from the second still 1, then 2, 3, 5, 8, 13, ...,
The sequence has many other features and even today many mathematicians try to find new properties connected to it. The Fibonacci sequence and the golden section
With the golden section indicates, usually, in art and mathematics the relationship between two unequal magnitudes of which the largest is the mean proportional between the child and their sum:
((a + b): a = a: b).
This ratio is approximately 1.618. Apparently an irrational number like everyone else, but its mathematics and geometry and the abundant presence in various natural settings have made a canon of harmony and beauty that has always attracted artists and intellectuals of all time.
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