Fibonacci numbers next to us

Fibonacci numbers surround us everywhere. They are in music, architecture, poetry, mathematics, economics, in the stock market, in the structure of plants, in the spiral of the snail, in the proportions of the human body and so on, to infinity ...

Famous medieval mathematician LeonardoPisansky (about 1170-about 1250), better known as Fibonacci, was one of the most famous scientists of his time. For the first time in Europe he proposed using Arabic numerals instead of Roman numerals and discovered a mathematical sequence of numbers, later named after him, which looks like this: 1,1,2,3,5,8,13,21, ... and so on ad infinitum. The sequence of these numbers is sometimes called the "Fibonacci numbers".

It is easy to see that in this remarkablesequence, each next number is formed as a result of the addition of the two previous ones. And what is it remarkable about? If we divide each next member of this unique sequence by the previous one, then we will gradually approach a certain amazing transcendental relation - the number of F (the Fibonacci number) = 1.6180339887 ...

This number, like the number Pi (3,1415 ...) does not havethe exact value. The number of digits after the decimal point is infinite. This is the beginning of mathematical and not only miracles. If we divide any member of the sequence into the next one, we also get a transcendental number of 0, 6180339887 ... Miracles continue - after the decimal, the digits exactly repeat the sequence of digits of the number Φ, only before the comma not 1, but 0.

Go ahead. If we square any Fibonacci number, the result will be the product of the number in the sequence before it multiplied by the number that stands behind it, plus or minus 1. For example, five squares equals 3x8 plus 1; 8 in the square is 5x13 minus 1; 13, squared, is equal to 8x21 plus 1 and so on. The signs "plus" and "minus" change, alternating. There are a lot of such mathematical miracles here. Fibonacci numbers work miracles around us, we just sometimes do not notice it.

Fibonacci numbers in nature

The Fibonacci ratios, bearing different names,Golden proportions, Golden section, Divine proportions - are found in the most unexpected and mysterious places. For example, these relationships can be met by careful consideration of the geometric proportions of the pyramid in Giza, the pyramids in Mexico, the monument of the ancient architecture of the Parthenon.

In plants, you can also see this magicalratio. We can observe the Fibonacci numbers again if we carefully consider the inflorescences of various composite plants: in the iris flower we find 3 petals, in primrose - 5, in ragweed ragweed - 13, in Leucanthemum vulgaris -34, and in asters - 55 and 89 petals .

Great Goethe noticed and studied the manifestationhelicity in nature. Spirals can be seen in the way the seeds of sunflower, pine cones, cacti, pineapples, etc. are located. In all these cases, the Fibonacci number is manifested. Spider spins its spider web spiraling. Hurricanes are twisted spirally. So the galaxies are twisted. "The curve of life" - the so-called Johann Goethe's spiral.

Finds its manifestation of the Fibonacci ratio inbiology of different organisms. For example, the number of rays of sea stars corresponds to the Fibonacci numbers. A simple mosquito can also be found: legs have 3 pairs, 8 segments have an abdomen, and on the head there are 5 antennas. The number of vertebrae in some animals is 55 and so on.

The lizard has a ratio of the length of its tail to the restthe length of the body is 62 and 38, and this ratio is harmonious and pleasing to our eye. In the animal and plant world, symmetry is everywhere. God, Nature or Great Architect has divided into symmetrical segments, parts and golden proportions. In part, the structure of the whole can be repeated, which is a manifestation of fractality in nature.

Gold symmetry is observed in transitions,associated with the energy costs of elementary particles, in the structure of individual chemical compounds, in space systems, in genetic structures, in the structure of certain organs of man and his body, is manifested in biorhythms, brain work, and perceptual properties.