Encyclopedia
In
mathematics, the
Fibonacci numbers form a sequence defined
recursively 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
...
,
2584, 4181, 6765, 10946, 17711,
The Fibonacci numbers are named after Leonardo of Pisa, known as
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 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 reviews this work in his
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 . 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 , 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:
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, the ratio of consecutive Fibonacci numbers, that is:
converges to the
golden ratio 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
eigenvalues of the matrix A are and , and the elements of the
eigenvectors 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. 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 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 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 .
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 Fibonacci numbers are sometimes used to determine tunings, and, as in visual art, to determine the length or size of
content or formal elements. Examples include
Béla Bartók's
Music for Strings, Percussion, and Celesta. \Since the conversion factor 1.609 for
miles to kilometers is close to the
golden mean 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.
Przemyslaw Prusinkiewicz has advanced the idea that these can be in part understood as the expression of certain algebraic constraints on
free groups, specifically as certain Lindenmayer grammars.
Identities
- F = F + F
- F + F + F + … + F = F − 1
- F + 2 F + 3 F + … + n F = n F − F + 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;
……
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. Suppose that
Fn and
Fn+1 have a common factor
g. Then
Fn−1 =
Fn+1 –
Fn must also be a multiple of
g; and by
induction 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=
FnLnIdentity for doubling n
Another identity useful for calculating F
n 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 constanthas been proved irrational by Richard André-Jeannin, but no closed form expression for it is known.
Generalizations
Extension to negative integers
Using F
n-2 = F
n - F
n-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 = -
nF
n.
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 . 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. This sequence was described by
Leonhard Euler in 1748, in the
Introductio in Analysin Infinitorum. The significance in the Lucas numbers
L lies in the fact that raising the
golden ratio 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 = PU − QU
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 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.
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
x4 −
x3 −
x2 −
x − 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 + 3
F.
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 algorithms.
Popular culture
Architecture
External links
...
,
Cornwall,
England.
Cinema
- Referenced in the film Dopo Mezzanotte where the sequence appears as neon numbers on the dome of the Mole Antonelliana in Turin
Turin is a major industrial city [i] in north-western Italy [i], capital [i] of the Piedmont [i] ...
,
Italy and is also used to select numbers in a lottery, ultimately winning it.
- Along with the concepts of the golden rectangle and golden spiral, the fibonacci sequence is used in Darren Aronofsky's independent film p
- Referenced in the film of The Phantom Tollbooth.
- It was also used as a key plot point in an episode of the Disney original television series 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
...
.
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
- 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.
- The sequence is used in the novel The Wright 3 by Blue Balliett.
- In Phillip K. Dick's novel VALIS, the Fibonnacci sequence are used as identification signs by an organization called the "Friends of God".
Music
- MC Paul Barman 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's song "Lateralus" 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
...
- Erno Lendvai analyzes Béla Bartók'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
...
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'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 MathNets episode "The Case of the Willing Parrot."
- The fibonacci sequence is also referenced to in Numb3rs, 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. 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 is Marilyn Manson [i]'s second full-length studio release and wa ...
in his depiction of "The Vitruvian Man", in the vein of Leonardo DaVinci'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
...
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
...
, 2005.
- in many programming languages.
-
- and - examples of how Fibonacci's work is used every day in the futures markets.
