ABSTRACT
The sunflower pattern is a botanic form whose structure realizes the principle of spiral and symmetrical structuring. On the pattern surface there is a clearly structured figure that consists of cross-sectional left- and right-handed spiral lines-parastichies formed by the seeds, which are blocked together. The quantity of left and right parastichies indicates the order of symmetry of the sunflower pattern and, as a rule, is described by two adjacent numbers of Fibonacci series (F-sequence)-recurrent sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,… which is formed according to the rule Fn-1+Fn=Fn+1, where Fn-n is the sequence member. It is typical that where , where is the golden section (proportion). It can also be noticed that , where D is the angle of the so-called divergence, which is the circular disconnection of two subsequent in terms of age in sunflower seeds. As, -1, then is the maximum value of D.
The classical model of spiral and symmetrical formation of pattern structure is based on the idea of the so-called locally isotropic growth when all the elements of the structure show equal growth preserving similarity. Consequently, parastichies are logarithmic spirals. The research provided in the book demonstrates different growth mechanisms. It is based on the so-called composite spiral, that is, the total pattern of simultaneous motion of the disk-growing element along the hyperbole and circle. The analytical and geometrical apparatus of our model is based on the so-called golden hyperbolic functions:
Golden cosine
Golden sine
Golden tangent, etc.
Consequently, golden functions can be used to represent Fibonacci series through these functions: Fn=2k-1= Gsh n; Fn+1 = 2k=Gch (n+1), where k = 1, 2, 3, …
Finally, the apparatus of the golden functions makes it possible to provide an exhaustive mathematical description of phyllotaxis growth process. Also new is the fact is proved that the growth mechanism of spiral and symmetrical vegetation forms implements Minkowski geometry.
