The Fibonacci sequence is a series of numbers in which each number is the sum of the previous two. It starts with 0 and 1, which equals 1. Then 1 plus 2 equals 3, 2 plus 3 equals 5, and so on.

n mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

With seed values[1]

The Fibonacci numbers are represented practically everywhere. In the petals on a flower, or the arrangement of leaves along a stem, you will find this sequence of numbers. The petals on most flowers display one of the Fibonacci numbers. The numbers also appear in certain parts of sea shell formations. Parts of the human body also reveal these ratios, including the five fingers, and a thumb on each hand. Fibonacci also can be seen in a piano that produces harmony through a beautiful music. A piano has one keyboard with five black keys (sharps and flats) arranged in groups of two and three, and eight white keys (whole tones) for the 13 chromatic musical octaves.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci.

IN NATURE

A model for the pattern of florets in the head of a sunflower was proposed by H. Vogel in 1979.[52] 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.[53]

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Occurrences in mathematics

The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle (see Binomial coefficient).[16]

The Fibonacci numbers can be found in different ways in the sequence of binary strings. * The number of binary strings of length n without consecutive 1s is the Fibonacci number Fn+2. For example, out of the 16 binary strings of length 4, there are F6 = 8 without consecutive 1s – they are 0000, 0100, 0010, 0001, 0101, 1000, 1010 and 1001. By symmetry, the number of strings of length n without consecutive 0s is also Fn+2. * The number of binary strings of length n without an odd number of consecutive 1s is the Fibonacci number Fn+1. For example, out of the 16 binary strings of length 4, there are F5 = 5 without an odd number of consecutive 1s – they are 0000, 0011, 0110, 1100, 1111. * The number of binary strings of length n without an even number of consecutive 0s or 1s is 2Fn. For example, out of the 16 binary strings of length 4, there are 2F4 = 6 without an even number of consecutive 0s or 1s – they are 0001, 1000, 1110, 0111, 0101, 1010.

FRACTALS

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension[1] and may fall between the integers.[2]Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far"[3] Fractals may be exactly the same at every scale, or as illustrated in Figure 1, they may be nearly the same at different scales.[2][4][5][6] The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.[2]:166; 18[4][7] As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional...