MAGIC OF FIBBONACSI IN FINANCIAL ASTROLOGY

Fibonacci number

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"Fibonacci Sequence" redirects here. For the chamber ensemble, see Fibonacci Sequence (ensemble).

A tiling with squares whose side lengths are successive Fibonacci numbers

In mathematics, the Fibonacci numbers, commonly denoted  F_n form a  sequence , called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,^[1]

${\displaystyle F_{0}=0,\quad F_{1}=1,}$

and

${\displaystyle F_{n}=F_{n-1}+F_{n-2},}$

for n > 1.

One has F₂ = 1. In some books, and particularly in old ones, F₀, the "0" is omitted, and the Fibonacci sequence starts with F₁ = F₂ = 1. The beginning of  the  sequence  is thus:

${\displaystyle (0,)\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots }$

The Fibonacci spiral: an approximation of the golden spiralcreated by drawing circular arcsconnecting the opposite corners of squares in the Fibonacci tiling;^[5] this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13 and 21.

Fibonacci numbers are strongly related to the

golden ratio: Binet's formulaexpresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases.

                          Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as  Fibonacci. They appear to have first arisen as early as 200 BC in work by  Pingala  on enumerating possible patterns of poetry formed from syllables of two lengths. In his 1202 book  Liber Abaci , Fibonacci introduced the sequence to Western European mathematics,

although the sequence had been described earlier in Indian mathematics.

                 Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the  Fibonacci search technique and the Fibonacci heap data structure, and graphs called  Fibonacci cubes used for interconnecting parallel and distributed systems.

                                   They also appear in biological settings, such as branching in trees ,

the arrangement of leaves on a stem , the fruit sprouts of a pineapple , the flowering of an artic hoke, an uncurling fern and the arrangement of a pine cone's bracts.

                               Fibonacci numbers are also closely related to  Lucas numbers ${\displaystyle L_{n}}$ in that they form a complementary pair of Lucas sequences ${\displaystyle U_{n}(1,-1)=F_{n}}$ and ${\displaystyle V_{n}(1,-1)=L_{n}}$. Lucas numbers are also intimately connected with the  golden ratio.