"Golden wurf"

Telling about the new models of the biological objects growth, it is necessary to mention the researches of the Russian biomechanics Sergey Petoukhov. He introduced the relevant concept of the "golden wurf" (from the German word of "wurf" that means a "spurt"); this concept got a broad resonance in the biological science.

It is known, that the growth of any biological object (including a man) is accompanied by the change of their configurations on the basis of the bio-symmetry laws. As it is shown the bio-symmetry researches made by Petoukhov, the conformal transformations lie in their basis. Let's remind, that the "conformal mapping" in the given point calls the continuous mapping keeping fixed the angles between curves passing through the given point. If the mapping is conformal in all points of some area, it is called by the conformal one in this area.

At the conformal transformations in the general case generally both sizes and forms of the body change. The conformal transformations have the relevant significance at the analysis of the age changes of proportions in the kinematical configuration of the man body. The kinematical configuration of the man and in general the backbone of animal consists of trinomial kinematical units: tri-phalanx fingers, tri-limbed finiteness (shoulder - forearm - hand, femur - shin - foot), tri-limbed body (upper, trunk and lower segments). Petoukhov showed that the proportions of the lengths of the tri-limbed man's body units during its growth change by the rules of the conformal transformations: "in each tri-limbed unit the growth lengthening of one link is matched to the lengthening of two other links so, that in the straightened unit there is saved an invariable invariant of one-dimensional circumferential (and design) transformations called as the double cross-ratio or "wurf":

where in brackets the corresponding lengths of the links between the four split points (A, B, C, D) of the tri-limbed unit are given".

The experimental researches showed that the values of the "wurfs" with adequate accuracy grouped around of the value of 1,31, which corresponds to the "wurf" of the hand finger. Let's find the connection of the indicated value of the "wurf" with the Fibonacci numbers. For this purpose we will consider the Fibonacci series

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..., Fn, Fn+1, Fn+2, ...

and let's make from this series the sequence of the ratios of the Fibonacci numbers taken through one:

Let's divide each number of this sequence by 2:

Now let's consider three adjacent numbers from the Fibonacci series Fn, Fn+1, Fn+2 and let's interpret these numbers as three line segments. Then let's find the "wurfs" of these three line segments sequence by using the general "wurf's" definition considered above:

Petoukhov called the limit of this sequence as the "golden wurf".

Thus, the mean "wurf's" value of the tri-limbed kinematical man's body units coincides with the value of the "golden wurf".

Note that Petoukhov's concept of the "wurf" proportion is principally new. In Petoukhov's opinion, it is possible to consider the "golden wurf" as the tri-limbed canon of the human body, and the system of the aesthetic proportions construction as the system of the "golden wurf".