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