# Fibonacci and nature

Plants have no way to know of Fibonacci numbers, but they develop in the most efficient way. Thus, many plants have leaf arrangement ordered in a Fibonacci sequence around strain. Some pine cones observe a disposition date of Fibonacci numbers, and also a sunflower. The rings on the trunks of palm trees meet Fibonacci numbers. The reason for all this is an optimum, a maximum effective. Astfelm for example, following Fibonacci sequence of leaves of plants can be arranged so as to occupy a smaller space and get a lot more sun.

The idea of leaf arrangement in this regard to consider leaving the angle of 222 gold, 5 degrees, the angle divided by the entire 360 degrees will result in the number 0.61803398 …, known as a string of Fibonacci rations.

In another vein, the petals of flowers is often a number of the Fibonacci sequence:

-Iris, lily: 3 petaled

- Dogberry, viorele, tulips, most flowers: 5 petaled

- Margaretele may have 34 or 21 petaled petaled most common

and examples are countless. Flowers should be noted that a number of petalous which are not Fibonacci sequence sunrise considered rare and special: hold (a leaf), Euphorbia (2 petaled), etc. …

Snail shell

How many of you have studied a little snail shell out "to walk" after a summer rain. Its design follows a highly successful spiral, a spiral that new we would be hard to achieve trasand with a pen. Having studied in detail in May, concluded that the spiral follows the data sizes of Fibonacci sequence:

-Focus on the positive: 1, 2, 5, 13, etc. …

- Focus on the negative: 0, 1, 3, 8, etc. ..

As you can see, these 2 subsiruri combined, will give even numbers of Fibonacci.

Reason and motivation for this layout is simple: in this way created ii snail shell, inside a maximum of space and safety. It is still one of various examples of application of sequence in nature.

Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers are also found in the family tree of honeybees.

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979. This has the form

where n is the index number of the floret and c is a constant scaling factor; the florets thus lie on Fermat's spiral. The divergence angle, approximately 137.51°, is the golden angle, dividing the circle in the golden ratio. Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently. Because the rational approximations to the golden ratio are of the form F(j):F(j + 1), the nearest neighbors of floret number n are those at n ± F(j) for some index j which depends on r, the distance from the center. It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other (or some other pair of adjacent Fibonacci numbers), but this is true only of one range of radii, typically the outermost and thus most conspicuous.