In Depth
See Also

Fibonacci number

In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, the Fibonacci numbers form a sequence defined recursively Recursion

In mathematics [i] and computer science [i], recursion specifies a class of objects or methods by defi ... 

 by: In other words, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers for n = 0, 1, are The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci Fibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Fibonacci, or simply Fibonacci... 

, although they had been described earlier in India.

Discussions

  Discussion Features

   Ask a question about 'Fibonacci number'

   Start a new discussion about 'Fibonacci number'

   Answer questions about 'Fibonacci number'

   'Fibonacci number' discussion forum


Encyclopedia

In mathematics Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ... 

, the Fibonacci numbers form a sequence defined recursively Recursion

In mathematics [i] and computer science [i], recursion specifies a class of objects or methods by defi... 

 by:

In other words, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers for n = 0, 1, … are
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 1000

1000 was a leap year starting on Monday [i].

... 

, 2584 2000

2000 was a leap year starting on Saturday [i] of the Gregorian calendar [i]. ... 

, 4181, 6765, 10946, 17711,



The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci Fibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Fibonacci, or simply Fibonacci... 

, although they had been described earlier in India.

Origins

The "Fibonacci" numbers first appear, under the name maatraameru , in the work of the Sanskrit grammarian Pingala . Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka  showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain Jainism

Jainism , traditionally known as Jain Dharma , is a religion [i] and philosophy [i] originating in ... 

 philosopher Hemachandra  composed a well known text on these. A commentary on Virahanka by Gopala in the 12th c. also revisits the problem in some detail.

Sanskrit vowel sounds can be long or short , and Virahanka's analysis, which came to be known as mAtrA-vritta wishes to compute how many metres of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are:

1 mora: S
2 morae: SS; L
3 morae: SSS, SL; LS
4 morae: SSSS, SSL, SLS; LSS, LL
5 morae: SSSSS, SSSL, SSLS, SLSS, SLL; LSSS, LSL, LLS


A pattern of length n can be formed by adding S to a pattern of length n-1, or L to a pattern of length n-2; and the prosodicists showed that the number of patterns of length n is the sum of the two previous numbers in the series. Donald Knuth Donald Knuth

Donald Ervin Knuth is a renowned computer scientist [i] and [i] ... 

 reviews this work in his The Art of Computer Programming The Art of Computer Programming

The Art of Computer Programmingidered an expert at writing compilers [i], Knuth started to write a b ... 

as equivalent formulations of the problem of bin packing items of length 1 and 2.

In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci Fibonacci

Leonardo of Pisa , also known as Leonardo Pisano, Leonardo Fibonacci, or simply Fibonacci... 

 . He considers the growth of an idealised rabbit population, assuming that:

  • in the first month there is just one newly-born pair,
  • new-born pairs become fertile from their second month on
  • each month every fertile pair begets a new pair, and
  • the rabbits never die


Let the population at month n be F. At this time, only rabbits who
were alive at month n-2 are fertile and produce offspring, so F pairs are added to the current population of F. Thus the total is F = F + F.

The bee ancestry code


Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci sequence. One can derive this from the following sequence:

  • If an egg is laid by a single female, it hatches a male.
  • If, however, the egg is fertilized by a male, it hatches a female.
  • Thus, a male bee will always have one parent, and a female bee will have two.


If one traces the ancestry of this male bee , he has 1 female parent . This female had 2 parents, a male and a female . The female had two parents, a male and a female, and the male had one female . Those two females each had two parents, and the male had one . If one continues this sequence, it gives a perfectly accurate depiction of the Fibonacci sequence.

Notice that this is a mathematical statement, it does not describe actual bee ancestries. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents.

Relation to the golden ratio


The golden ratio Golden ratio

The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

  , is defined as the ratio that results when a line is divided so that the whole line has the same ratio to the larger segment as the larger segment has to the smaller segment. Expressed mathematically, normalising the larger part to unit length, it is the positive solution of the equation:

or equivalently

which is equal to .

Closed form expression

Like every sequence defined by linear recursion, the Fibonacci numbers have a closed-form solution. It has become known as Binet's formula:

, where is the golden ratio defined above.

Note the similarity between the Fibonacci recursion:

to the defining equation of the golden ratio in the form

also known as the generating polynomial of the recursion.

Proof :

Any root of the equation above satifies and multiplying by shows:

Note that by definition is a root of the equation and that the other root is . Therefore:

= and
=



Now consider the functions:

defined for any real

All these functions satisfy the Fibonacci recursion

 
 
 
 


Selecting and gives the formula of Binet we started with. It has been shown that this formula satisfies the Fibonacci recursion. Furthermore:

and

establishing the base cases of the induction, proving that

for all n.

Note that for any two starting values, a combination can be found such that the function is the exact closed formula for the series.

In fact, since for all , is the closest integer to .
For computational purposes, this is expressed using the floor function,

Limit of consecutive quotients


As was pointed out by Johannes Kepler Johannes Kepler

Johannes Kepler , a key figure in the scientific revolution [i], was a German [i] mathematician [i] ... 

, the ratio of consecutive Fibonacci numbers, that is:

converges to the golden ratio Golden ratio

The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

 

Actually, this limit behaviour does not depend on the particular starting values .

Proof:

It follows from the explicit formula that for any real :
  
 
,



because, as is easily shown, and thus

Matrix form


A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

or

The eigenvalue Eigenvalue, eigenvector and eigenspace

In mathematics [i], an of a transformation [i]In this context, only linear transformation [i] ... 

s of the matrix A are and , and the elements of the eigenvector Eigenvalue, eigenvector and eigenspace

In mathematics [i], an of a transformation [i]In this context, only linear transformation [i] ... 

s of A, and , are in the ratios and .

Note that this matrix has a determinant of −1, and thus it is a 2×2 unimodular matrix Unimodular matrix

In mathematics [i], a unimodular matrix is a square matrix [i] with determinant [i] +1 or − ... 

. This property can be understood in terms of the continued fraction representation for the golden mean:  = [1; 1, 1, 1, 1, …]. The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for , and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.

The matrix representation gives the following closed expression for the Fibonacci numbers:

Taking the determinant of both sides of this equation yields the identity

Additionally, since for any square matrix , the following identities can be derived:


Applications


The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm Euclidean algorithm

In number theory [i], the Euclidean algorithm is an algorithm [i] to determine the greatest common divisor [i] ... 

 to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers.

Yuri Matiyasevich Yuri Matiyasevich

Yuri Matiyasevich, is a Russian [i] mathematician [i]. ... 

 was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.

The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle Pascal's triangle

In mathematics [i], Pascal's triangle is a geometric arrangement of the binomial coefficient [i]s in a triangle [i]... 

 .

Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation.

Fibonacci numbers are used by some pseudorandom number generators.

A one-dimensional optimization method, called the Fibonacci search technique uses Fibonacci numbers .

In music Music

Music is an art, entertainment [i], or other human activity that involves organized and audible sounds a ... 

 Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of content Content

Content can mean:
As an adjective:
... 

 or formal elements. Examples include Béla Bartók Béla Bartók

Bla Viktor Jnos Bartk was a Hungarian [i] composer [i], pianist [i] and collector of Eastern Europe [i] ... 

's Music for Strings, Percussion, and Celesta. \Since the conversion factor 1.609 for mile Mile

[i], usually used to measure [[distance]... 

s to kilometers is close to the golden mean Golden ratio

The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

 f, 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 base f being shifted. To go from kilometers to miles shift the register down the Fibonacci sequence instead.

Fibonacci numbers in nature


Fibonacci sequences have been noted to appear in biological settings, such as branching in trees and the arrangement of a pine cone Pine Cone

Sorry, no overview for this topic 

. Przemyslaw Prusinkiewicz Przemyslaw Prusinkiewicz

Przemyslaw Prusinkiewicz is a Polish [i] mathematician [i] who advanced the idea that Fibonacci number [i] ... 

 has advanced the idea that these can be in part understood as the expression of certain algebraic constraints on free group Free group

In mathematics [i], a group [i] G is called free if there is a subset [i] S of G such ... 

s, specifically as certain Lindenmayer grammars.

Identities

F = F + F


F + F + F + … + F = F − 1


F + 2 F + 3 F + … + n F = n FF + 2


F2 + F2 + F2 + … + F2 = F F


These identities can be proven using many different methods.
But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here.
In particular, F can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F = 0, meaning no sum will add up to −1, and that F = 1, meaning the empty sum will "add up" to 0.
Here the order of the summands matters.
For example, 1 + 2 and 2 + 1 are considered two different sums and are counted twice.

Proof of the first identity.
Without loss of generality, we may assume n = 1.
Then F counts the number of ways summing 1's and 2's to n.

When the first summand is 1, there are F ways to complete the counting for n − 1; and the first summand is 2, there are F ways to complete the counting for n − 2.
Thus, in total, there are F + F ways to complete the counting for n.

Proof of the second identity.
We count the number of ways summing 1's and 2's to n + 1 such that at least one of the summands is 2.

As before, there are F ways summing 1's and 2's to n + 1 when n = 0.
Since there is only one sum of n + 1 that does not use any 2, namely 1 + … + 1 , we subtract 1 from F.

Equivalently, we can consider the first occurrence of 2 as a summand.
If, in a sum, the first summand is 2, then there are F ways to the complete the counting for n − 1.
If the second summand is 2 but the first is 1, then there are F ways to complete the counting for n − 2.
Proceed in this fashion.
Eventually we consider the th summand.
If it is 2 but all of the previous n summands are 1's, then there are F ways to complete the counting for 0.
If a sum contains 2 as a summand, the first occurrence of such summand must take place in between the first and th position.
Thus F + F + … + F gives the desired counting.

Proof of the third identity.
This identity can be established in two stages.
First, we count the number of ways summing 1s and 2s to −1, 0, …, or n + 1 such that at least one of the summands is 2.

By our second identity, there are F − 1 ways summing to n + 1; F − 1 ways summing to n; …; and, eventually, F − 1 way summing to 1.
As F − 1 = F = 0, we can add up all n + 1 sums and apply the second identity again to obtain
   [F − 1] + [F − 1] + … + [F − 1]
= [F − 1] + [F − 1] + … + [F − 1] + [F − 1] + F
= F + [F + … + F + F] −
= F + F − .


On the other hand, we observe from the second identity that there are
  • F + F + … + F + F ways summing to n + 1;
  • F + F + … + F ways summing to n;

……
  • F way summing to −1.

Adding up all n + 1 sums, we see that there are
  • F + n F + … + F ways summing to −1, 0, …, or n + 1.


Since the two methods of counting refer to the same number, we have
F + n F + … + F = F + F


Finally, we complete the proof by subtracting the above identity from n + 1 times the second identity.

Common factors


Any two consecutive Fibonacci numbers are relatively prime Coprime

In mathematics [i], the integer [i]s a and b are said to be coprime or relatively prime if ... 

. Suppose that Fn and Fn+1 have a common factor g. Then Fn−1 = Fn+1Fn must also be a multiple of g; and by induction Mathematical induction

Mathematical induction is a method of mathematical proof [i] typically used to establish that a given st ... 

 the same must be true of all lower Fibonacci numbers. But F1 = 1, so g = 1.

Other identities include relationships to the Lucas numbers, which have the same recursive properties but start with L0=2 and L1=1. These properties include
F2n=FnLn

Identity for doubling n

Another identity useful for calculating Fn for large values of n is

for all integers n and k.

Power series

The Fibonacci power series

has a simple and interesting closed-form solution for x < 1/f:

This function is therefore the generating function of the Fibonacci sequence. It can be proven as follows:
Substituting :

  
  
  
  
  
  



Therefore,

In particular, math puzzle-books note the curious value . The sum is easily proved by noting that

and then explicitly evaluating the sum.

Reciprocal sums


Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions.

The reciprocal Fibonacci constant

has been proved irrational by Richard André-Jeannin, but no closed form expression for it is known.

Generalizations


Extension to negative integers

Using Fn-2 = Fn - Fn-1, one can extend the Fibonacci numbers to negative integers. So we get: ... -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, ... and F-n = -nFn.

Vector space

The term Fibonacci sequence is also applied more generally to any function g from the integers to a field for which g = g + g. These functions are precisely those of the form g = F'g + F'g, so the Fibonacci sequences form a vector space with the functions F and F as a basis.

More generally, the range of g may be taken to be any abelian group Abelian group

In mathematics [i], an abelian group, also called a commutative group, is a group [i] such ... 

 . Then the Fibonacci sequences form a 2-dimensional Z-module in the same way.

Similar integer sequences


Lucas numbers
In particular, the Fibonacci sequence L with L = 1 and L = 3 is referred to as the Lucas numbers, after Edouard Lucas Edouard Lucas

Franois douard Anatole Lucas was a French [i] mathematician [i]. ... 

. This sequence was described by Leonhard Euler Leonhard Euler

Leonhard Euler was a Swiss [i] mathematician [i] and physicist [i]. ... 

 in 1748, in the Introductio in Analysin Infinitorum. The significance in the Lucas numbers L lies in the fact that raising the golden ratio Golden ratio

The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

 to the nth power yields




Lucas numbers are related to Fibonacci numbers by the relation

A generalization of the Fibonacci sequence are the Lucas sequences. One kind can be defined thus:

U = 0
U = 1
U = PUQU


where the normal Fibonacci sequence is the special case of P = 1 and Q = −1. Another kind of Lucas sequence begins with V = 2, V = P. Such sequences have applications in number theory and primality proving.

The Padovan sequence Padovan sequence

The Padovan sequence is the sequence [i] of integer [i]s P(n) defined by the initial values
... 

 is generated by the recurrence P = P + P.
Tribonacci numbers
The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are :
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, …


The tribonacci constant is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x3 − x2 − x − 1, approximately 1.83929, and also satisfies the equation x + x−3 = 2. It is important in the study of the snub cube Snub cube

The snub cube, or snub cuboctahedron, is an Archimedean solid [i].
... 

.

The tribonacci numbers are also given by

where the outer brackets denote the nearest integer function and

.
Tetranacci numbers
The tetranacci numbers start with four predetermined terms, each term afterwards being the sum of the preceding four terms. The first few tetranacci numbers are :
0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, …


The tetranacci constant is the ratio toward which adjacent tetranacci numbers tend. It is a root of the polynomial x4x3x2x − 1, approximately 1.92756, and also satisfies the equation x + x−4 = 2.
Other -anacci numbers
Pentanacci, hexanacci and heptanacci numbers have been computed, with perhaps less interest so far in research.

Interestingly, there is a limit to this with increasing n. A 'polynacci' sequence, if one could be described, would after an infinite number of zeroes yield the sequence [..., 0, 0, 1,] 1, 2, 4, 8, 16, 32, ... which are simply powers of 2.

Other generalizations

The Fibonacci polynomials are another generalization of Fibonacci numbers.

A random Fibonacci sequence can be defined by tossing a coin for each position n of the sequence and taking F=F+F if it lands heads and F=F-F if it lands tails. Work by Furstenburg and Kesten guarantees that this sequence almost surely grows exponentially at a constant rate: the constant is independent of the coin tosses and was computed in 1999 by Divakar Viswanath. It is now known as Viswanath's constant.

A repfigit or Keith number is an integer, that when its digits start a Fibonacci sequence with that number of digits, the original number is eventually reached. An example is 47, because the Fibonacci sequence starting with 4 and 7 reaches 47. A repfigit can be a tribonacci sequence if there are 3 digits in the number, a tetranacci number if the number has four digits, etc. The first few repfigits are :

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, …


Since the set of sequences satisfying the relation S = S + S is closed under termwise addition and under termwise multiplication by a constant, it can be viewed as a vector space. Any such sequence is uniquely determined by a choice of two elements, so the vector space is two-dimensional. If we abbreviate such a sequence as , the Fibonacci sequence F = and the shifted Fibonacci sequence F = are seen to form a canonical basis for this space, yielding the identity:

S = S'F + S'F


for all such sequences S. For example, if S is the Lucas sequence 1, 3, 4, 7, 11…, then we obtain L = F + 3F.

Fibonacci primes


The first few Fibonacci numbers that are also prime numbers are : 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …. It is not known if there are infinitely many Fibonacci primes.

Fibonacci strings


In analogy to its numerical counterpart, a Fibonacci string is defined by:
,
where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts:

b, a, ab, aba, abaab, abaababa, abaababaabaab, …


The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number.

Fibonacci strings appear as inputs for the worst case in some computer algorithm Algorithm

In mathematics [i] and computing [i], an algorithm is a procedure for accomplishing some task which, gi... 

s.

Popular culture


Architecture

  • The sequence has been used in the design of a building, the Core, at the Eden Project Eden Project

    The Eden Project is a large-scale environmental [i] complex near St Austell [i], Cornwall [i],... 

    , near St Austell St Austell

    |colspan=2 align=center|

External links
... 

, Cornwall Cornwall

Cornwall is a county [i] in South West [i]... 

, England England

England is the largest and most populous constituent country [i] of the United Kingdom [i]. ... 

.

Cinema

  • Referenced in the film Dopo Mezzanotte where the sequence appears as neon numbers on the dome of the Mole Antonelliana Mole Antonelliana

    The Mole Antonelliana is a major landmark [i] of the Italian [i] city Turin [i].... 

     in Turin Turin



Turin is a major industrial city [i] in north-western Italy [i], capital [i] of the Piedmont [i] ... 

, Italy Italy

Italy, officially the Italian Republic , is a Southern European [i] country. ... 

 and is also used to select numbers in a lottery, ultimately winning it.
  • Along with the concepts of the golden rectangle Golden rectangle

    A golden rectangle is a rectangle [i] whose side lengths are in the golden ratio [i], 1:φ, that is, ... 

     and golden spiral Golden spiral

    In geometry [i], a golden spiral is a logarithmic spiral [i] whose growth factor b is related to &ph ... 

    , the fibonacci sequence is used in Darren Aronofsky Darren Aronofsky

    Darren Aronofsky is an American [i] film director [i], screenwriter [i] and film producer [i]... 

    's independent film p Pi

    The mathematical constant [i] p is an irrational [i] real number [i], approximately eq ... 

  • Referenced in the film of The Phantom Tollbooth.
  • It was also used as a key plot point in an episode of the Disney The Walt Disney Company

    The Walt Disney Company is one of the largest media and entertainment corporations in the world.... 

     original television series So Weird So Weird

    so weird was a television series [i] shot in Vancouver [i] that aired on the Disney Channel [i] from ... 

    .
  • Used in Steven Spielbergs miniseries Taken Taken

    * For the band, see Taken [i]

... 

.
  • In The Da Vinci Code The Da Vinci Code

    The Da Vinci Code is a mystery [i]/detective [i] novel [i] by American [i] ... 

     an encrypted message was firstly thought to represent the Fibonacci numbers.

Literature

  • The sequnce is mentioned in the book "Raising Atlantis" by Thomas Greanias.
  • The fibonacci sequence plays a small part in the bestselling novel and film The Da Vinci Code The Da Vinci Code

    The Da Vinci Code is a mystery [i]/detective [i] novel [i] by American [i] ... 

  • Fibs  have been popularized by Gregory K. Pincus on his blog, .
  • A part of the fibonacci sequence is used as a code in Matthew Reilly's novel Ice Station Ice Station (novel)

    Ice Station is Australia [i]n thriller [i] writer Matthew Reilly [i]'s second novel, ... 

    .
  • The sequence is used in the novel The Wright 3 The Wright 3

    The Wright 3 is the sequel to Chasing Vermeer [i], by Blue Balliett [i] and illustrated by Brett Helquist [i] ... 

     by Blue Balliett.
  • In Phillip K. Dick Philip K. Dick

    Philip Kindred Dick was an American [i] science fiction [i] writer [i]. ... 

    's novel VALIS, the Fibonnacci sequence are used as identification signs by an organization called the "Friends of God".

Music

  • MC Paul Barman MC Paul Barman

    MC Paul Barman is a hip hop [i] MC [i] from Ridgewood, New Jersey [i], who attended ... 

     structured the rhymes in his song "Enter Pan-Man" according to the fibonacci sequence.
  • Dr. Steel released a song titled "Fibonacci Sequence" in 2005.
  • BT released a dance track in 2000, entitled the "Fibonacci Sequence," which features a sample of a reading of the sequence.
  • Tool Tool

    A tool or device is a piece of equipment that provides a mechanical advantage [i] in accomp ... 

    's song "Lateralus Lateralus

    Lateralus is the third full-length album from the band Tool [i]. It was released in May 2001. ... 

    " from the album of the same name features the fibonacci sequence symbolically in the verses of the song. The syllables in the first verse count 1, 1, 2, 3, 5, 8, 5, 3, 13, 8, 5, 3. Similarly, on Tool's 10,000 Days album there has already been speculation to more fibonacci references embedded within the album.
  • The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence Fibonacci number

    In mathematics [i], the Fibonacci numbers form a sequence [i] defined recursively [i] by:

... 


  • Erno Lendvai  analyzes Béla Bartók Béla Bartók

    Bla Viktor Jnos Bartk was a Hungarian [i] composer [i], pianist [i] and collector of Eastern Europe [i] ... 

    's works as being based on two opposing systems, that of the golden ratio and the acoustic scale. In Bartok's Music for Strings, Percussion and Celeste the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.
  • French composer Erik Satie Erik Satie

    Eric Alfred Leslie Satie was a French [i] composer [i], pianist [i] and writer [i].

... 

 used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.
  • The Fibonacci numbers are also apparent in the organisation of the sections in the music of Debussy Claude Debussy

    Achille-Claude Debussy was a French [i] composer [i]. ... 

    's Image, Reflections in Water, in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8.Television


  • The fibonacci sequence is a key plot point in the television show MathNet Mathnet

    Mathnet, a segment on Square One [i] and spoof [i] of Dragnet [i], feature ... 

    s episode "The Case of the Willing Parrot."
  • The fibonacci sequence is also referenced to in Numb3rs NUMB3RS

    Numbers is an American [i] television show [i] that follows FBI [i] ... 

    , the television series. Many times the cast reference note the relationship the sequence has with nature to further emphasise the wonders of mathematics.


Visual Arts


  • In a FoxTrot comic, Jason and Marcus are playing football American football

    American football, known in the United States [i] and Canada [i] simply as football, is a competit ... 

    . Jason yells, "Hut 0! Hut 1! Hut 1! Hut 2!" all the way until "Hut 13!" in the fibonacci sequence. Marcus yells, "Is it the fibonacci sequence?" Jason says, "Correct! Touchdown, Marcus!"
  • Marilyn Manson is another artist who has employed the Fibonacci sequence. He uses the sequence overtly in a watercolor painting entitled Fibonacci during his Holy Wood era, which it should be noted, uses bees as focal points. More discreetly, Manson used the sequence in the interior album art of Antichrist Superstar Antichrist Superstar

    Antichrist Superstar is Marilyn Manson [i]'s second full-length studio release and wa ... 

    in his depiction of "The Vitruvian Man Vitruvian Man

    The Vitruvian Man is a famous drawing [i] with accompanying notes by Leonardo da Vinci [i] made ar ... 

    ", in the vein of Leonardo DaVinci Leonardo da Vinci

    Leonardo di ser Piero da Vinci was a talented Italian Renaissance [i] Roman Catholic [i] ... 

    's work which was also based on the sequence. There is also that some of the beats in the songs on the album Holy Wood are based on the Fibonacci sequence as well.
  • Mario Merz frequently uses the fibonacci sequence in his art work

See also


  • Fibonacci number program
  • Anti-Fibonacci numbers
  • Golden ratio Golden ratio

    The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the ... 

  • Logarithmic spiral Logarithmic spiral

    A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral [i] curve [i] ... 

  • Plastic number
  • Padovan number Padovan sequence

    The Padovan sequence is the sequence [i] of integer [i]s P(n) defined by the initial values

... 


  • Perrin number

References


External links

  • — an academic journal devoted to the study of Fibonacci numbers
  • Alexey Stakhov, , .
  • Subhash Kak, , Archive of Physics, .
  • Ron Knott, , .
  • Ron Knott, , .
  • Bob Johnson, ,
  • Donald E. Simanek, , .
  • Rachel Hall, , .
  • Schweizer, David. , .
  • Alex Vinokur, , .
  • , , .
  • - Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, the Golden section and the Golden string.
  • The incorporated in 1963, focuses on Fibonacci numbers and related mathematics, emphasizing new results, research proposals, challenging problems, and new proofs of old ideas.
  • Dawson Merrill's link page.
  • - Rachael Hall surveys rhythm and Fibonacci numbers and also the Hemachandra connection. Saint Joseph's University Saint Joseph's University

    name = Saint Joseph's University

... 

, 2005.
  • in many programming languages.
  • and - examples of how Fibonacci's work is used every day in the futures markets.





Categories: