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AEFD rectangle with the parties of AE=φAD is called as a gold rectangle. ABCD quadrangle - a square. It is easy to see that a rectangle of BEFC also gold, as BC=a =φ. This circumstance suggests at once an idea of further splitting a rectangle of BEFC.

The creation of the correct pentagon offered by Euclid includes division of a piece of a straight line on average and the extreme relation, called subsequently a golden ratio and drawing to itself attention of artists and architects throughout several centuries.

Regular polygons drew attention of Ancient Greek scientists of da Archimedes still long before. The Pythagoreans who chose as an emblem of the union a pentagram - a five-pointed star, attached very great value to a task about division of a circle into equal parts, that is about creation of the regular entered polygon. Albrecht Duerer (1471-1527g, become the Renaissance embodiment in Germany gives theoretically exact way of creation of the correct pentagon borrowed from the great composition of Ptolemaeus of "Almagest".

At division of any member of sequence Fibonacci on following it simply return turns out to the 618th size (1: 618=6 Ho it too very unusual, even remarkable phenomenon. As the initial ratio – infinite fraction, at this ratio also should not be the end.

more clear if to show the attitudes of the several first members of sequence. In this example the relations of the second member to the first, third to the second, fourth are brought to the third, and so on:

Medical properties of pyramids, especially golden ratio are widely known. In some most popular beliefs, the room in which there is such pyramid, seems more, and air - is more transparent. Dreams start being remembered better. It is also known that a golden ratio it was widely applied in architecture and a sculpture. An example of that became: A pantheon and Parthenon in Greece, buildings of architects Bazhenov and Malevich

This sequence asymptotically (coming nearer all more slowly and the medleena aspires to some constant ratio. However, this ratio is irrational, that is represents number with infinite, unpredictable sequence of decimal figures in fractional part. It cannot be expressed precisely.

From a point In the perpendicular equal to a half of AV is restored. The received point With connects the line to a point And. On the received line VS piece which is coming to an end with a point of D is postponed. The piece of AD is transferred to direct AV. The point received thus E divides AV piece in the ratio of a gold proportion.

Ioann Kepler living five centuries ago possesses the statement: "The geometry possesses two great treasures. The first is Pythagorean theorem, the second - divisions of a piece in the extreme and average relation"

In the next centuries the rule of a gold proportion turned into the academic canon and when over time in art fight against the academic routine began, in the heat of fight "together with water splashed out also the child". The golden ratio in the middle of the XIX century was "reopened". In 1855 the German researcher of a golden ratio professor Tseyzing published the work "Aesthetic Researches". To Tseyzing there was that, as it had to is inevitable to happen to the researcher who considers the phenomenon as that, without communication with other phenomena. It absolutized a proportion of a golden ratio, having declared its universal for all natural phenomena and art. Tseyzinga had numerous followers, but there were also opponents who declared its doctrine proportions "a mathematical esthetics".

Division is carried out as follows. The piece of AV shares in a proportion of a golden ratio. From a point With CD perpendicular vosstavlyatsya. The point of D which connects the line to a point is the radius of AV And. The right angle of ACD is halved. From a point With a line before crossing with the AD line is drawn. The point E divides AD piece concerning 56: 4