It only at a superficial glance can seem that the mathematics is boring. And that it is from beginning to end invented by the person for the needs: to count, to calculate, to draw properly. But if to dig more deeply, then, it appears, the abstract science reflects the natural phenomena. So, a set of objects of the terrestrial nature and all Universe can be described through the sequence of numbers of Fibonacci and also the principle of "golden ratio" connected with it.
What is Fibonacci's sequence
Fibonacci's sequence call a numerical row in which the first two numbers are equal 1 and 1 (option: 0 and 1), and each following number is the sum of two previous.
That definition became more clear, look how numbers for the sequence are chosen:
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5
- 3 + 5 = 8
- 5 + 8 = 13
And so as much as long. As a result the sequence looks so:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, etc.
For the ignorant person these numbers look only as result of a chain of additions, no more than that. But not everything is so simple.
As Fibonacci removed the well-known row
The sequence bears a name of the Italian mathematician Fibonacci (a real name - Leonardo Pizanscy) who lived the 12-13th centuries. He was not the first person who found this number sequence: before it already used in Ancient India. But pizanets opened the sequence for Europe.
The focus of interest of Leonardo Pizanscy included drawing up and the solution of tasks. One of them was about reproduction of rabbits.
Such conditions:
- on an ideal farm behind a fence there live rabbits and never die;
- originally animal two: male and samochka;
- on the second and each next month of the life of steam gives rise new (a rabbit plus a doe-rabbit);
- each new couple in the same way from second month of existence produces new couple, etc.
Task question: how many couples of animals will be on a farm in a year?
If to carry out calculations, then the number of rabbit couples will grow so:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.
That is, their quantity will increase according to the sequence described above.
Fibonacci's number and number F
But application of numbers of Fibonacci was not limited to the solution of a task about rabbits. It became clear that at the sequence there are a lot of remarkable properties. The most known consists in the attitudes of numbers of a row towards the previous values.
Let's consider one after another. With division of unit into unit (the result is equal to 1), and then the two on unit (private 2) everything is clear. But further results of division of the next members are at each other very curious:
- 3: 2 = 1.5
- 5: 3 = 1.667 (in round figures)
- 8: 5 = 1.6
- 13: 8 = 1.625
- ...
- 233: 144 = 1.618 (in round figures)
The result of division of any number of Fibonacci on previous (except very first) is close to so-called number F (fi) = 1.618. And the more a dividend and a divider, the closer private to this unusual number.
And than it, number F, is remarkable?
Number F expresses the relation of two sizes an and b (a thus more, than b) when equality is fair:
a/b = (a+b)/a.
That is, numbers in this equality have to be picked up so that division and into b yielded the same result, as well as division of the sum of these numbers on and. And always this result will be 1.618.
Strictly speaking, is 1.618 a rounding. A fractional part of number F lasts indefinitely as it is irrational fraction. Here so it looks with the first ten figures after a comma:
Ф = 1.6180339887
In the percentage of number and and b make about 62% and 38% of their sum.
When using a similar ratio in creation of figures forms harmonious and pleasant to a human eye turn out. Therefore the ratio of sizes which at division bigger on smaller give number F is called "golden ratio". Number F is called as "gold number".
It turns out that Fibonacci's rabbits bred in a "gold" proportion!
The term "golden ratio" is often connected with Leonardo da Vinci. Actually, the great artist and the scientist though applied this principle in the works, did not use such formulation. The name for the first time was in writing recorded much later - in the 19th century, in works of the German mathematician Martin Om.
Fibonacci's spiral and spiral of "golden ratio"
On the basis of Fibonacci's numbers and "golden ratio" it is possible to construct spirals. Sometimes these two figures are identified, but to speak about two different spirals more precisely.
Fibonacci's spiral is built so:
- draw two squares (one party the general), length of the parties is equal 1 (centimeter, inch or a cage - it is unimportant). The rectangle divided in two which long party is equal to 2 turns out;
- to the long party of a rectangle add a square with the party 2. The image of the rectangle divided into several parts turns out. Its long party is equal to 3;
- process is continued as much as long. At the same time new squares "attach" in a row only on or only counterclockwise;
- in the very first small square (with the party 1) draw from a corner to a corner a circle quarter. Then without interruption draw the similar line in each following square.
As a result receive a beautiful spiral which radius constantly and in proportion increases.
The spiral of "golden ratio" is drawn on the contrary:
- build "a gold rectangle" which parties correspond in the proportion of the same name;
- allocate a square which parties are equal to the short party of "a gold rectangle" in a rectangle;
- at the same time in a big rectangle there will be a square and a rectangle less. That, in turn, too will be "golden";
- the small rectangle is divided by the same principle;
- process is continued as much as long, having each new square turbinal;
- in small squares draw the quarters of a circle connected among themselves.
So the logarithmic spiral which grows according to golden ratio turns out.
Fibonacci's spiral and "gold" are very similar. But there is the main difference: the figure constructed on the sequence of the Pisa mathematician has an initial point though final - no. And here the "gold" spiral twists "inside" to infinitesimal numbers, as well as is untwisted "in out of" to infinitely big.
Examples of application
If the term "golden ratio" is rather new, then the principle was known from antiquity. Including, it is applied during creation of such world famous cultural objects:
- Egyptian pyramid of Cheops (about 2600 BC)
- Ancient Greek temple Parthenon (5th century BC)
- Leonardo da Vinci's works. The most striking example - Mona Lisa (beginning of the 16th century).
Use of "golden ratio" - one of answers to a riddle why the listed works of art and architecture seem to us fine.
"Golden ratio" and Fibonacci's sequence formed the basis of the best works of painting, architecture, a sculpture. And not only. So, Johann Sebastian Bach used it in some of the pieces of music.
Fibonacci's numbers were useful even in the financial sphere. They are used by the traders trading on share and the foreign exchange markets.
"Golden ratio" and Fibonacci's numbers in the nature
But why we so admire works of art in which "golden ratio" is applied? The answer is simple: this proportion is set by the nature.
Let's return to Fibonacci's spiral. Spirals of many mollusks are quite so twirled. For example, nautilus.
We meet similar spirals also in flora. For example, broccoli inflorescences romanesko and a sunflower and also a pine cone are so formed.
The structure of spiral galaxies corresponds to Fibonacci's spiral too. Let's remind that ours also treats such galaxies - the Milky Way. And also one of the next to us - Andromeda's Galaxy.
Fibonacci's sequence is also reflected in arrangement of leaves and branches at different plants. To numbers of a row there corresponds the quantity of flowers, petals in many inflorescences. Lengths of phalanxes of human fingers correspond approximately as Fibonacci's numbers too - or as pieces in "golden ratio".
In general, it is necessary to tell about the person separately. We consider beautiful those faces which parts precisely correspond to proportions of "golden ratio". Figures are perceived well built if parts of a body correspond by the same principle.
The structure of bodies of many animals is combined with this rule too.
Similar examples move some people to a thought that "golden ratio" and Fibonacci's sequence are the cornerstone of the universe. As if all: both the person, and the environment surrounding him and all Universe correspond to these principles. It is possible that in the future the person will find new proofs of a hypothesis and will manage to create convincing mathematical model of the world.