Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following sequence:

By definition, the first two Fibonacci numbers are 0 and 1, and each remaining number is the sum of the previous two. Some sources omit the initial 0, instead beginning the sequence with two 1s.

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

with seed values

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.[2][3]

List of Fibonacci numbers

The first 21 Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, 2, ... ,20 are:[9][10]

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Using the recurrence relation, the sequence can also be extended to negative index n. The result satisfies the equation

Thus the complete sequence is

Golden phi

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.[1] Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.[2][3][4] Other terms encountered include extreme and mean ratio,[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[6] golden number, and mean of Phidias.[7][8][9] The golden ratio is often denoted by the Greek letter phi, usually lower case (φ).

The figure on the right illustrates the geometric relationship that defines this constant. Expressed

algebraically:

This equation has as its unique positive solution the algebraic irrational number

[1]

1) Numeric definition

Here is a 'Fibonacci series'.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..

If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers.

1/1 = 12/1 = 23/2 = 1.55/3 = 1.6666...8/5 = 1.613/8 = 1.62521/13 = 1.61538...34/21 = 1.61904...

The ratio seems to be settling down to a particular value, which we call the golden ratio(Phi=1.618..).

List of numbers – Irrational and suspected irrational numbers

γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δS – α – e – π – δ

Binary 1.1001111000110111011…

Decimal 1.6180339887498948482…

Hexadecimal 1.9E3779B97F4A7C15F39…

Continued fraction

Algebraic form

Infinite series

A golden rectangle is one whose side lengths are in the golden ratio, 1: (one-to-phi), that is,

or approximately 1:1.618.

A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle; that is, with the same proportions as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property.

3) The Golden Spiral

The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle.If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi.

Applications

1 The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.[

2 Use in forex trading and stock markets

Human expectations occur in a ratio that approaches Phi

Changes in stock prices largely reflect human opinions, valuations and expectations.  A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative.

Phi and Fibonacci numbers are used to predict stocks

Phi (1.618), the Golden Mean and the numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, ...) have been used with great success to analyze and predict stock market moves.  Forbes ASAP featured a story on the work of scientist Stephen Wolfram in cellular automata (underlying rules that determine seemingly random phenomenon) stating "This seashell may hold the secret of stock market behavior, computers that think and the future of science."

3 The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods e.g. µ-law.[3

4 Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio (denoted φ), the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio base φ being shifted. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead

5 Art and architecture

The Great Pyramid

 

The Ahmes papyrus of Egypt gives an account of the building of the Great Pyramid of Giaz in 4700 B.C. with proportions according to a "sacred ratio."

 

2) Parthenon

The Greek sculptor Phidias sculpted many things including the bands of sculpture that run above the columns of the Parthenon.

Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion" has been known since it occurs naturally in some of the proportions of the Five Platonic. This rectangle is supposed to appear in many of the proportions of that famous ancient Greek temple in the Acropolis in Athens, Greece.

) Mona-Risa by Leonardo Da Vinci

This picture includes lots of Golden Rectangles. In above figure, we can draw a rectangle whose base extends from the woman's right wrist to her left elbow and extend the rectangle vertically until it reaches the very top of her head. Then we will have a golden rectangle. Also, if we draw squares inside this Golden Rectangle, we will discover that the edges of these new squares come to all the important focal points of the woman: her chin, her eye, her nose, and the upturned corner of her mysterious mouth. It is believed that Leonardo, as a mathematician tried to incorporate of mathematics into art. This painting seems to be made purposefully line up with golden rectangle.

Daily life

Petals on flowers

On many plants, the number of petals is a Fibonacci number:buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.The links here are to various flower and plant catalogues:

the Dutch Flowerweb's searchable index called Flowerbase.

The US Department of Agriculture's Plants Database containing over 1000 images, plant information and searchable database.

Fuchsia Pinks Lily

Daisies available as

a poster at

AllPosters.com

3 petals: lily, iris       Mark Taylor (Australia), a grower of Hemerocallis and Liliums (lilies) points out that although these appear to have 6 petals as shown above, 3 are in fact sepals and 3 are petals. Sepals form the outer protection of the flower when in bud. Mark's Barossa Daylilies web site (opens in a new window) contains many flower pictures where the

difference between sepals and petals is clearly visible. 4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia above, do. 4 is not a Fibonacci number! We return to this point near the bottom of this page.5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks (shown above)       The humble buttercup has been bred into a multi-petalled form. 8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria, some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family. Some species are very precise about the number of petals they have - e.g. buttercups, but others have petals that are very near those above, with the average being a Fibonacci numbe

Leaf arrangements

Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem.Here's a computer-generated image, based on an African violet type of plant, whereas this has lots of leaves.

Leaves per turn

The Fibonacci numbers occur when counting both the number of times we go around the stem, going from leaf to leaf, as well as counting the leaves we meet until we encounter a leaf directly above the starting one.

If we count in the other direction, we get a different number of turns for the same number of leaves.

The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers!

For example, in the top plant in the picture above, we have 3 clockwise rotations before we meet a leaf directly above the first, passing 5 leaves on the way. If we go anti-clockwise, we need only 2 turns. Notice that 2, 3 and 5 are consecutive Fibonacci numbers.For the lower plant in the picture, we have 5 clockwise rotations passing 8 leaves, or just 3 rotations in the anti-clockwise direction. This time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence.We can write this as, for the top plant, 3/5 clockwise rotations per leaf ( or 2/5 for the anticlockwise direction). For the second plant it is 5/8 of a turn per leaf (or 3/8).

The sunflower here when viewed from the top shows the same pattern. It is the same plant whose side view is above. Starting at the leaf marked "X", we find the next lower leaf turning clockwise. Numbering the leaves produces the patterns shown here on the right.

Vegetables and Fruit

Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in outline. Looking carefully, you can see a centre point, where the florets are smallest. Look again, and you will see the florets are organised in spirals around this centre in both directions.