Law of spiral symmetry transformation

As is well known from biology a relative arrangement of very different sprouts arising in the cones of shoots is characterized by the "spiral symmetry". This arrangement principle was named "phyllotaxis". On the surface of phyllotaxis forms, especially in the closely packed botanic structures (pine cone, pineapple, cactus, head of sunflower etc.), one can see clearly visible left- and right curved series of sprouts. As to the symmetry order of phyllotaxis forms there exists a practice to indicate it through the ratios of the numbers corresponding to the number of the left- and right-hand spirals. In accordance with the law of phyllotaxis such ratios are given by the number sequence generated by the Fibonacci recurrent relationship

 Gn = Gn-1 + Gn-2. (1)

The most widespread types of phyllotaxis are those described through the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ... , the Lucas numbers 1, 3, 4, 7, 11, 18, ... or the number sequence 4, 5, 9, 14, 23, ... satisfying to the general recurrent formula (1).

It is well known that the process of the collective fruit growing is accompanied at the certain stage by a modification of the spiral symmetry order. As this takes place the modification is strictly regular and corresponds to the general rule of constructing the recurrent number sequences generated by (1). In the case of Fibonacci's phyllotaxis the progress of symmetry order is presented through the sequence:

 1:2 => 2:3 => 3:5 => 5:8 => 8:13 => 13:21 => ... . (2)

The change of the symmetry orders of phyllotaxis objects in accordance with (2) is called the dynamic symmetry.

A remarkable illustration of the dynamic symmetry is given by the fact of a regular difference of the spiral symmetry orders in the sunflower heads located on different levels of one and the same stem.

The spiral numbers in the sunflower discs are in direct dependence on their "age", i.e. the "older" disc corresponds to the bigger Fibonacci numbers. Most often the symmetry order of discs belonging to the same stem is characterized by the ratios of the Fibonacci numbers: 13:21, 21:34, 34:55, 55:89.

It is these all data that constitute the essence of the universally known "puzzle of phyllotaxis". A number of scientists, investigating this problem, assume that the phenomenon of phyllotaxis is of fundamental interdisciplinary importance. In the opinion of the famous Russian scientist V. Vernadsky, the problem of the biology symmetry is the key problem of biological science.

A new solution of the phyllotaxis problem was given recently by the Ukrainian scientists O. Bodnar in his book "Golden Section and Non-Euclidean Geometry in Nature and Art" (1994).

The essence of Bodnar's discovery consists of the following. Let's present the phyllotaxis object (for example the pine cone) as a cylinder (Fig. 1-a) expanded on the plane in the form of the phyllotaxis lattice (Fig. 1-b).

Then the dynamic symmetry of the phyllotaxis object may be simulated as a sequential interchange of the neighboring phyllotaxis lattices corresponding to the same phyllotaxis object and arising at the different stages of phyllotaxis object growth (Fig. 2).

Analysis of the phyllotaxis lattices in Fig.2 showed that this geometric transformation is simulated ideally with the help of the hyperbolic rotation, which is the basic geometric transformation of the hyperbolic geometry. It is the first key scientific result obtained by Oleg Bodnar.

Figure 1. The phyllotaxis cylinder and its phyllotaxis lattice.

Bodnar developed the new geometric theory of phyllotaxis based on the concept of the "golden" hyperbolic functions, which coincide with the hyperbolic Fibonacci functions. Bodnar proved that the passage of the phyllotaxis object from one stage to other is described by using the hyperbolic Fibonacci functions.

Bodnar's discovery is nothing but just the deciphering of the natural symmetry transformation mechanism in the spiral-symmetric (phyllotaxis) biostructures. Bodnar proved that the change of the spiral symmetry order is realized through the hyperbolic rotation that is the basic transformation of the hyperbolic geometry. At that the mathematical relationships of the hyperbolic Fibonacci's functions underlie the animated nature. Bodnar's theory is a very good explanation why the Fibonacci and Lucas numbers appears in the phyllotaxis objects.

Figure 2. Analysis of the dynamic symmetry of the phyllotaxis object.

As is well known the space-time properties of the special theory of relativity (Minkovsky's geometry) are given with the help of the hyperbolic geometry. That is why Bodnar's discovery reveals the immediate connection between the space-temporal, i.e. the fundamental laws of the animated and inanimate nature and testifies the unity of these laws. It follows from this discovery that the hyperbolic geometry is the universal geometrical model of the space-temporal processes taking place both in animate and inanimate nature. At that the fundamental transformation principle of natural processes is nothing but the hyperbolic rotation. However, there exists an essential distinction between the hyperbolic geometry of animate and inanimate nature. The hyperbolic geometry of the inanimate nature is based on the classical hyperbolic functions, the essence of which is the e-number being one of the most important numerical constants of mathematics. The hyperbolic geometry of the animate nature is based on the hyperbolic Fibonacci and Lucas functions, which essence is expressed with the Golden Section, the fundamental numerical constant of the animate nature.