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Gold relation

In the book ( " What is Mathematics " by Richard Courant and Herbert Robbins, Oxford University Press, London, New York, Toronto ) marked: " in opinion of the Greek mathematicians rectangle parties of that are in that behalf, esthetically especially pleasant for an eye ". This relation was adopted " gold ".

The indicated relation behaves not only to the rectangle but also to a straight limit line, in this relation. Such coincidence of description of objects, where a gold relation " is executed " can to be it contingently only fundamental properties of construction of numerical space. In these fundamental properties without fail must functional dependences will be executed. Fundamental figures in mathematics are a circle, triangle, straight line that give in the combination decisions of great number of tasks. Unusual properties show up at implementation of transformation relatively that is named a circumference inversion.

This transformation of the special points of triangle, located in circumferences, gives in the numerical great number of plane out of circumference a "gold relation". Circular functions in transformation in relation to a circumference determine appearance functional dependences " of gold relation ".

Choice of size of counting out of circular function equal 1/60 produced on the factors of coincidence of size of arc of circumference and sizes of chord, and the size of segment of circle is equal to 1/10 000 at the radius of circle of R = 1 .

Accepting such unit of counting out of circular function 1/60 sizes of circle as argument further procedure of calculation of function that is a circular size and is executed a corresponding to it size of chord : is a base size (1/60), there is a doubling further [2* (1/60)] - to this size, doubling of increase will be added up further - [2* (1/60)]+ 2* (1/60); [2* (1/60)]+ 2* (2* (1/60)) - (a chord is 0.618 at R=1); [2* (1/60)]+2* (2*2* (1/60)) - (a chord is 1.0 at R=1 ); [2* (1/60)]+2* (2*2*2* (1/60)) - ( a chord is 1.618 at R=1 or size of founding of isosceles triangle).

Fale (img002.jpg). Accepted as an argument circular size (1/60) changes on procedure the brought calculation over and the result of it is a size of chords circle at R= a 1 relation of that is "gold".

The gold relation algebraically registers in a look: a/b = b / (a + b) or b*b = a (a + b). In inversion the basic r radius circle, the center of inversion of O, point on the plane l and a point in a basic circle of k being on one beam are connected by dependence: Ok * Ol = r*r or Ok * (Ok + kl) = r*r.

Algebraic form of record of the gold relation and transformations of inversion are identical: [A] * ([A + B]) = [C*C] of or b*b, or r*r. The gold relation in transformation inversion is carried out at an arrangement on a beam of points of the geometrical figures characterizing identical qualities, for example special points of triangles which size of the parties is chosen according to the specified change of circular function. It confirms a conclusion that the geometrical constructions executed in a circle taking into account the harmonious changes of circular function correspond to geometrical constructions on the plane and they form the gold relation. Thus harmony of proportions of a circle is transferred to the plane or space. This phenomenon is one of universe bases.

I thank you,

Sergey Molchanov

e-mail: egypy@yandex.ua

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