Fibonacci sequence: Difference between revisions

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[[File:Fibonacci Squares.svg|thumb|A tiling with [[square]]s whose side lengths are successive Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21]]

The Fibonacci numbers were first described in [[Indian mathematics]] as early as 200 &nbsp;BC in work by [[Pingala]] on enumerating possible patterns of [[Sanskrit]] poetry formed from syllables of two lengths.<ref name="GlobalScience" /><ref name="HistoriaMathematica" /><ref name="Donald Knuth 2006 50" /> They are named after the Italian mathematician Leonardo of Pisa, also known as [[Fibonacci]], who introduced the sequence to Western European mathematics in his 1202 book {{lang|la|[[Liber Abaci]]}}.{{Sfn|Sigler|2002|pp=404–05}}

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 [[graph (discrete mathematics)|graphs]] called [[Fibonacci cube]]s used for interconnecting parallel and distributed systems. They also appear [[Patterns in nature#Spirals|in biological settings]], such as branching in trees, [[phyllotaxis|the arrangement of leaves on a stem]], the fruit sprouts of a [[pineapple]], the flowering of an [[artichoke]], and the arrangement of a [[pine cone]]'s bracts, though they do not occur in all species.

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The Fibonacci sequence appears in [[Indian mathematics]], in connection with [[Sanskrit prosody]].<ref name="HistoriaMathematica">{{Citation|first=Parmanand|last=Singh|title= The So-called Fibonacci numbers in ancient and medieval India|journal=Historia Mathematica|volume=12|issue=3|pages=229–244|year=1985|doi = 10.1016/0315-0860(85)90021-7|doi-access=free}}</ref><ref name="knuth-v1">{{Citation|title=The Art of Computer Programming|volume=1|first=Donald|last=Knuth| author-link =Donald Knuth |publisher=Addison Wesley|year=1968|isbn=978-81-7758-754-8|url=https://books.google.com/books?id=MooMkK6ERuYC&pg=PA100|page=100|quote=Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns&nbsp;... both Gopala (before 1135&nbsp;AD) and Hemachandra (c.&nbsp;1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)&nbsp;...}}</ref>{{sfn|Livio|2003|p=197}} In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration {{mvar|m}} units is {{math|''F''_''m''+1}}.<ref name="Donald Knuth 2006 50">{{Citation|title = The Art of Computer Programming | volume = 4. Generating All Trees – History of Combinatorial Generation | first = Donald | last = Knuth | author-link = Donald Knuth |publisher= Addison–Wesley |year= 2006 | isbn= 978-0-321-33570-8 | page = 50 | url= https://books.google.com/books?id=56LNfE2QGtYC&q=rhythms&pg=PA50 | quote = it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when {{math|1=''m'' = 7}} are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)}}</ref>

Knowledge of the Fibonacci sequence was expressed as early as [[Pingala]] ({{circa}}&nbsp;450&nbsp;BC–200&nbsp;BC). Singh cites Pingala's cryptic formula ''misrau cha'' ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for {{mvar|m}} beats ({{math|''F''_''m''+1}}) is obtained by adding one [S] to the {{math|''F''_''m''}} cases and one [L] to the {{math|''F''_''m''−1}} cases.<ref>{{Citation | last = Agrawala | first = VS | year = 1969 | title = ''Pāṇinikālīna Bhāratavarṣa'' (Hn.). Varanasi-I: TheChowkhamba Vidyabhawan | quote = SadgurushiShya writes that Pingala was a younger brother of Pāṇini [Agrawala 1969, lb]. There is an alternative opinion that he was a maternal uncle of Pāṇini [Vinayasagar 1965, Preface, 121]. ... Agrawala [1969, 463–76], after a careful investigation, in which he considered the views of earlier scholars, has concluded that Pāṇini lived between 480 and 410 BC}}</ref> [[Bharata Muni]] also expresses knowledge of the sequence in the ''[[Natya Shastra]]'' (c.&nbsp;100&nbsp;BC–c.&nbsp;350&nbsp;AD).<ref name="HistoriaMathematica"/><ref name=GlobalScience>{{Citation|title=Toward a Global Science|first=Susantha|last=Goonatilake|author-link=Susantha Goonatilake|publisher=Indiana University Press|year=1998|page=126|isbn=978-0-253-33388-9|url=https://books.google.com/books?id=SI5ip95BbgEC&pg=PA126}}</ref><ref name="HistoriaMathematica"/>

However, the clearest exposition of the sequence arises in the work of [[Virahanka]] (c.&nbsp;700 &nbsp;AD), whose own work is lost, but is available in a quotation by Gopala (c.&nbsp;1135):{{sfn|Livio|2003|p=197}}

<blockquote>Variations of two earlier meters [is the variation]&nbsp;... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21]&nbsp;... In this way, the process should be followed in all ''mātrā-vṛttas'' [prosodic combinations].{{efn|"For four, variations of meters of two [and] three being mixed, five happens. For five, variations of two earlier—three [and] four, being mixed, eight is obtained. In this way, for six, [variations] of four [and] of five being mixed, thirteen happens. And like that, variations of two earlier meters being mixed, seven [[Mora (linguistics)|morae]] [is] twenty-one. In this way, the process should be followed in all mātrā-vṛttas" <ref>{{Citation|last=Velankar|first=HD|year=1962|title='Vṛttajātisamuccaya' of kavi Virahanka|publisher=Rajasthan Oriental Research Institute|___location=Jodhpur|page=101}}</ref>}}</blockquote>

[[Hemachandra]] (c.&nbsp;1150) is credited with knowledge of the sequence as well,<ref name=GlobalScience/> writing that "the sum of the last and the one before the last is the number&nbsp;... of the next mātrā-vṛtta."{{sfn|Livio|2003|p=197–198}}<ref>{{citation|last1=Shah|first1=Jayant|year=1991|title=A History of Piṅgala's Combinatorics|url=https://web.northeastern.edu/shah/papers/Pingala.pdf|publisher=[[Northeastern University]]|page=41|access-date=4 January 2019-01-04}}</ref>

===Europe===

[[File:Liber abbaci magliab f124r.jpg|thumb|upright=1.25|A page of [[Fibonacci]]'s {{lang|la|[[Liber Abaci]]}} from the [[National Central Library (Florence)|Biblioteca Nazionale di Firenze]] showing (in box on right) 13 entries of the Fibonacci sequence:<br /> the indices from present to XII (months) as Latin ordinals and Roman numerals and the numbers (of rabbit pairs) as Hindu-Arabic numerals starting with 1, 2, 3, 5 and ending with 377.]]

The Fibonacci sequence first appears in the book {{lang|la|[[Liber Abaci]]}} (''The Book of Calculation'', 1202) by [[Fibonacci]],{{Sfn|Sigler|2002|pp=404–405}}<ref>{{citation|url=https://www.math.utah.edu/~beebe/software/java/fibonacci/liber-abaci.html|title=Fibonacci's Liber Abaci (Book of Calculation)|date=13 December 2009|website=[[The University of Utah]]|access-date=28 November 2018-11-28}}</ref> where it is used to calculate the growth of rabbit populations.<ref>{{citation

| last = Tassone | first = Ann Dominic

| date = April 1967

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* At the end of the fourth month, the original pair has produced yet another new pair, and the pair born two months ago also produces their first pair, making 5 pairs.

At the end of the {{mvar|n}}-th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month {{math|''n'' – 2}}) plus the number of pairs alive last month (month {{math|''n'' – 1}}). The number in the {{mvar|n}}-th month is the {{mvar|n}}-th Fibonacci number.<ref>{{citation | last = Knott | first = Ron

| title = Fibonacci's Rabbits | url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits | publisher =[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref>

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<math display=block>F_n=\left\lfloor\frac{\varphi^n}{\sqrt 5}\right\rceil,\ n \geq 0.</math>

In fact, the rounding error quickly becomes very small as {{mvar|n}} grows, being less than 0.1 for {{math|''n'' ≥ 4}}, and less than 0.01 for {{math|''n'' ≥ 8}}. This formula is easily inverted to find an index of a Fibonacci number {{mvar|F}}:

<math display=block>n(F) = \left\lfloor \log_\varphi \sqrt{5}F\right\rceil,\ F \geq 1.</math>

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=== Limit of consecutive quotients ===

[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio <math>{{tmath|\varphi\colon </math> }}:<ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>

<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>

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=== Identification ===

Binet's formula provides a proof that a positive integer {{mvar|x}} is a Fibonacci number [[if and only if]] at least one of <math>5x^2+4</math> or <math>5x^2-4</math> is a [[Square number|perfect square]].<ref>{{Citation | title = Fibonacci is a Square | last1 = Gessel | first1 = Ira | journal = [[The Fibonacci Quarterly]] | volume = 10 | issue = 4 | pages = 417–19 |date=October 1972 | url = https://www.fq.math.ca/Scanned/10-4/advanced10-4.pdf | access-date = April 11, 2012-04-11 }}</ref> This is because Binet's formula, which can be written as <math>F_n = (\varphi^n - (-1)^n \varphi^{-n}) / \sqrt{5}</math>, can be multiplied by <math>\sqrt{5} \varphi^n</math> and solved as a [[quadratic equation]] in <math>\varphi^n</math> via the [[quadratic formula]]:

<math display=block>\varphi^n = \frac{F_n\sqrt{5} \pm \sqrt{5{F_n}^{\!2} + 4(-1)^n}}{2}.</math>

Comparing this to <math>\varphi^n = F_n \varphi + F_{n-1} = (F_n\sqrt{5} + F_n + 2 F_{n-1})/2</math>, it follows that

: <math display=block>5{F_n}^{\!2} + 4(-1)^n = (F_n + 2F_{n-1})^2\,.</math>

In particular, the left-hand side is a perfect square.

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<math display=block> \vec F_{k+1} = \mathbf{A} \vec F_{k},</math>

which yields <math>\vec F_n = \mathbf{A}^n \vec F_0</math>. The [[eigenvalue]]s of the [[matrix (mathematics)|matrix]] {{math|'''A'''}} are <math>\varphi=\tfrac12\bigl(1+\sqrt5~\!\bigr)</math> and <math>\psi=-\varphi^{-1}=\tfrac12\bigl(1-\sqrt5~\!\bigr)</math> corresponding to the respective [[eigenvector]]s

<math display=block>\vec \mu=\begin{pmatrix} \varphi \\ 1 \end{pmatrix}, \quad \vec\nu=\begin{pmatrix} -\varphi^{-1} \\ 1 \end{pmatrix}.</math>

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\end{align}</math>

where

<math display=block>

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<math display=block>\varphi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}.</math>

The [[convergent (continued fraction)|convergents]] of the continued fraction for {{mvar|φ}} are ratios of successive Fibonacci numbers: {{math|1=''φ''_''n'' = ''F''_''n''+1 / ''F''_''n''}} is the {{mvar|n}}-th convergent, and the {{math|(''n''&thinsp;+&thinsp;1)}}-st convergent can be found from the recurrence relation {{math|1=''φ''_''n''+1 = 1 + 1 / ''φ''_''n''}}.<ref>{{Cite web |title=The Golden Ratio, Fibonacci Numbers and Continued Fractions. |url=https://nrich.maths.org/2737 |access-date=2024-03-22 |website=nrich.maths.org |language=en}}</ref> The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1. The matrix representation gives the following closed-form expression for the Fibonacci numbers:

<math display=block>\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.</math>

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Following the same logic as before, by summing the cardinality of each set we see that

: <math>F_{n+2} = F_n + F_{n-1} + ... + |\{(1,1,...,1,2)\}| + |\{(1,1,...,1)\}|</math>

... where the last two terms have the value <math>F_1 = 1</math>. From this it follows that <math>\sum_{i=1}^n F_i = F_{n+2}-1</math>.

A similar argument, grouping the sums by the position of the first&nbsp;1 rather than the first&nbsp;2 gives two more identities:

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A different trick may be used to prove

<math display=block>\sum_{i=1}^n F_i^2 = F_n F_{n+1}</math>

or in words, the sum of the squares of the first Fibonacci numbers up to <math>F_n</math> is the product of the {{mvar|n}}-th and {{math|(''n'' + 1)}}-th Fibonacci numbers. To see this, begin with a Fibonacci rectangle of size <math>F_n \times F_{n+1}</math> and decompose it into squares of size <math>F_n, F_{n-1}, ..., F_1</math>; from this the identity follows by comparing [[area]]sareas:

[[File:Fibonacci Squares.svg|frameless|260x260px]]

=== Symbolic method ===

The sequence <math>(F_n)_{n\in\mathbb N}</math> is also considered using the [[symbolic method (combinatorics)|symbolic method]].<ref>{{citation |last1=Flajolet |first1=Philippe |last2=Sedgewick |first2=Robert |title=Analytic Combinatorics|title-link= Analytic Combinatorics |date=2009 |publisher=Cambridge University Press |isbn=978-0521898065 |page=42}}</ref> More precisely, this sequence corresponds to a [[specifiable combinatorial class]]. The specification of this sequence is <math>\operatorname{Seq}(\mathcal{Z+Z^2})</math>. Indeed, as stated above, the <math>n</math>-th Fibonacci number equals the number of [[Composition (combinatorics)|combinatorial compositions]] (ordered [[integer partition|partitions]]) of <math>n-1</math> using terms 1 and 2.

It follows that the [[ordinary generating function]] of the Fibonacci sequence, <math>\sum_{i=0}^\infty F_iz^i</math>, is the [[rational function]] <math>\frac{z}{1-z-z^2}.</math>

=== Induction proofs ===

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<math display=block>

s(z) = \sum_{k=0}^\infty F_k z^k = 0 + z + z^2 + 2z^3 + 3z^4 + 5z^5 + \dotscdots.

</math>

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<math display=block>s(z) = s\!\left(-\frac{1}{z}\right).</math>

Using <math>z</math> equal to any of 0.01, 0.001, 0.0001, etc. lays out the first Fibonacci numbers in the decimal expansion of <math>s(z)</math>. For example, <math>s(0.001) = \frac{0.001}{0.998999} = \frac{1000}{998999} = 0.001001002003005008013021\ldots.</math>

== Reciprocal sums ==

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Every third number of the sequence is even (a multiple of <math>F_3=2</math>) and, more generally, every {{mvar|k}}-th number of the sequence is a multiple of ''F_k''. Thus the Fibonacci sequence is an example of a [[divisibility sequence]]. In fact, the Fibonacci sequence satisfies the stronger divisibility property<ref>{{Citation | first = Paulo | last = Ribenboim | author-link = Paulo Ribenboim | title = My Numbers, My Friends | publisher = Springer-Verlag | year = 2000}}</ref><ref>{{Citation | last1 = Su | first1 = Francis E | others = et al | publisher = HMC | url = http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | contribution = Fibonacci GCD's, please | year = 2000 | title = Mudd Math Fun Facts | access-date = 2007-02-23 | archive-url = https://web.archive.org/web/20091214092739/http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml | archive-date = 2009-12-14 | url-status = dead }}</ref>

<math display=block>\gcd(F_a,F_b,F_c,\ldots) = F_{\gcd(a,b,c,\ldots)}\,</math>

where {{math|gcd}} is the [[greatest common divisor]] function. (This relation is different if a different indexing convention is used, such as the one that starts the sequence with {{tmath|1=F_0 = 1}} and {{tmath|1=F_1 = 1}}.)

In particular, any three consecutive Fibonacci numbers are pairwise [[Coprime integers|coprime]] because both <math>F_1=1</math> and <math>F_2 = 1</math>. That is,

: <math>\gcd(F_n, F_{n+1}) = \gcd(F_n, F_{n+2}) = \gcd(F_{n+1}, F_{n+2}) = 1</math>

for every {{mvar|n}}.

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The above formula can be used as a [[primality test]] in the sense that if

<math display=block>n \mid F_{n \;-\, \left(\frac{5}{n}\right)},</math>

where the Legendre symbol has been replaced by the [[Jacobi symbol]], then this is evidence that {{mvar|n}} is a prime, and if it fails to hold, then {{mvar|n}} is definitely not a prime. If {{mvar|n}} is [[composite number|composite]] and satisfies the formula, then {{mvar|n}} is a ''Fibonacci pseudoprime''. When {{mvar|m}} is large{{snd}}say a 500-[[bit]] number{{snd}}then we can calculate {{math|''F''_''m'' (mod ''n'')}} efficiently using the matrix form. Thus

<math display=block> \begin{pmatrix} F_{m+1} & F_m \\ F_m & F_{m-1} \end{pmatrix} \equiv \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^m \pmod n.</math>

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<math display=block>F_1 = 1,\ F_3 = 2,\ F_5 = 5,\ F_7 = 13,\ F_9 = {\color{Red}34} = 2 \cdot 17,\ F_{11} = 89,\ F_{13} = 233,\ F_{15} = {\color{Red}610} = 2 \cdot 5 \cdot 61.</math>

All known factors of Fibonacci numbers {{math|''F''(''i'')}} for all {{math|''i'' < 50000}} are collected at the relevant repositories.<ref>{{Citation | url = https://mersennus.net/fibonacci/ | title = Fibonacci and Lucas factorizations | publisher = Mersennus}} collects all known factors of {{math|''F''(''i'')}} with {{math|''i'' < 10000}}.</ref><ref>{{Citation | url =http://fibonacci.redgolpe.com/ | title = Factors of Fibonacci and Lucas numbers | publisher = Red golpe}} collects all known factors of {{math|''F''(''i'')}} with {{math|10000 < ''i'' < 50000}}.</ref>

=== Periodicity modulo ''n'' ===

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* Generalizing the index to [[real number]]s using a modification of Binet's formula.<ref name="MathWorld" />

* Starting with other integers. [[Lucas number]]s have {{math|1=''L''₁ = 1}}, {{math|1=''L''₂ = 3}}, and {{math|1=''L_n'' = ''L''_''n''−1 + ''L''_''n''−2}}. [[Primefree sequence]]s use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite.

* Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The [[Pell number]]s have {{math|1=''P_n'' = 2''P''_''n''−1 + ''P''_''n''−2}}. If the coefficient of the preceding value is assigned a variable value {{mvar|x}}, the result is the sequence of [[Fibonacci polynomials]].

* Not adding the immediately preceding numbers. The [[Padovan sequence]] and [[Perrin number]]s have {{math|1=''P''(''n'') = ''P''(''n'' − 2) + ''P''(''n'' − 3)}}.

* Generating the next number by adding 3 numbers (tribonacci numbers), 4 numbers (tetranacci numbers), or more. The resulting sequences are known as ''n-Step Fibonacci numbers''.<ref>{{citation

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The Fibonacci numbers can be found in different ways among the set of [[binary numeral system|binary]] [[String (computer science)|strings]], or equivalently, among the [[subset]]s of a given set.

* The number of binary strings of length {{mvar|n}} without consecutive {{math|1}}s is the Fibonacci number {{math|''F''_''n''+2}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''₆ = 8}} without consecutive {{math|1}}s—they are 0000, 0001, 0010, 0100, 0101, 1000, 1001, and 1010. Such strings are the binary representations of [[Fibbinary number]]s. Equivalently, {{math|''F''_''n''+2}} is the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without consecutive integers, that is, those {{mvar|S}} for which {{math|{{mset|''i'', ''i'' + 1}} ⊈ ''S''}} for every {{mvar|i}}. A [[bijection]] with the sums to {{math|''n''+1}} is to replace 1 with 0 and 2 with 10, and drop the last zero.

* The number of binary strings of length {{mvar|n}} without an odd number of consecutive {{math|1}}s is the Fibonacci number {{math|''F''_''n''+1}}. For example, out of the 16 binary strings of length 4, there are {{math|1=''F''₅ = 5}} without an odd number of consecutive {{math|1}}s—they are 0000, 0011, 0110, 1100, 1111. Equivalently, the number of subsets {{mvar|S}} of {{math|{{mset|1, ..., ''n''}}}} without an odd number of consecutive integers is {{math|''F''_''n''+1}}. A bijection with the sums to {{mvar|n}} is to replace 1 with 0 and 2 with 11.

* The number of binary strings of length {{mvar|n}} without an even number of consecutive {{math|0}}s or {{math|1}}s is {{math|2''F''_''n''}}. For example, out of the 16 binary strings of length 4, there are {{math|1=2''F''₄ = 6}} without an even number of consecutive {{math|0}}s or {{math|1}}s—they are 0001, 0111, 0101, 1000, 1010, 1110. There is an equivalent statement about subsets.

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* A one-dimensional optimization method, called the [[Fibonacci search technique]], uses Fibonacci numbers.<ref>{{Citation| first1 = M | last1 = Avriel | first2 = DJ | last2 = Wilde | title= Optimality of the Symmetric Fibonacci Search Technique |journal=Fibonacci Quarterly|year=1966 |issue=3 |pages= 265–69| doi = 10.1080/00150517.1966.12431364 }}</ref>

* The Fibonacci number series is used for optional [[lossy compression]] in the [[Interchange File Format|IFF]] [[8SVX]] audio file format used on [[Amiga]] computers. The number series [[companding|compands]] the original audio wave similar to logarithmic methods such as [[μ-law]].<ref>{{Citation | title = Amiga ROM Kernel Reference Manual | publisher = Addison–Wesley | year = 1991}}</ref><ref>{{Citation | url = https://wiki.multimedia.cx/index.php?title=IFF#Fibonacci_Delta_Compression | contribution = IFF | title = Multimedia Wiki}}</ref>

* Some Agile teams use a modified series called the "Modified Fibonacci Series" in [[planning poker]], as an estimation tool. Planning Poker is a formal part of the [[Scaled agile framework|Scaled Agile Framework]].<ref>{{citation|author=Dean Leffingwell |url=https://www.scaledagileframework.com/story/ |title=Story |publisher=Scaled Agile Framework |date=2021-07-01 |accessdateaccess-date=2022-08-15}}</ref>

* [[Fibonacci coding]]

* [[Negafibonacci coding]]

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[[File:FibonacciChamomile.PNG|thumb|[[Yellow chamomile]] head showing the arrangement in 21 (blue) and 13 (cyan) spirals. Such arrangements involving consecutive Fibonacci numbers appear in a wide variety of plants.]]

Fibonacci sequences appear in biological settings,<ref>{{Citation |first1=S |last1=Douady |first2=Y |last2=Couder |title=Phyllotaxis as a Dynamical Self Organizing Process |journal=Journal of Theoretical Biology |year=1996 |issue=3 |pages=255–74 |url=http://www.math.ntnu.no/~jarlet/Douady96.pdf |doi=10.1006/jtbi.1996.0026 |volume=178 |url-status=dead |archive-url=https://web.archive.org/web/20060526054108/http://www.math.ntnu.no/~jarlet/Douady96.pdf |archive-date=2006-05-26 }}</ref> such as branching in trees, [[Phyllotaxis|arrangement of leaves on a stem]], the fruitlets of a [[pineapple]],<ref>{{Citation | first1=Judy |last1=Jones | first2=William | last2=Wilson |title=An Incomplete Education |publisher=Ballantine Books |year=2006 |isbn=978-0-7394-7582-9 |page=544 |chapter=Science}}</ref> the flowering of [[artichoke]], the arrangement of a [[pine cone]],<ref>{{Citation| first=A | last=Brousseau |title=Fibonacci Statistics in Conifers | journal=[[Fibonacci Quarterly]] |year=1969 |issue=7 |pages=525–32}}</ref> and the family tree of [[honeybee]]s.<ref>{{citation|url = https://www.cs4fn.org/maths/bee-davinci.php |work = Maths | publisher = Computer Science For Fun: CS4FN |title = Marks for the da Vinci Code: B–}}</ref><ref>{{Citation|first1=T.C.|last1=Scott|first2=P.|last2=Marketos| url = http://www-history.mcs.st-andrews.ac.uk/Publications/fibonacci.pdf | title = On the Origin of the Fibonacci Sequence | publisher = [[MacTutor History of Mathematics archive]], University of St Andrews| date = March 2014}}</ref> [[Kepler]] pointed out the presence of the Fibonacci sequence in nature, using it to explain the ([[golden ratio]]-related) [[pentagon]]al form of some flowers.{{sfn|Livio|2003|p=110}} Field [[Leucanthemum vulgare|daisies]] most often have petals in counts of Fibonacci numbers.{{sfn|Livio|2003|pp=112–13}} In 1830, [[Karl Friedrich Schimper]] and [[Alexander Braun]] discovered that the [[Parastichy|parastichies]] (spiral [[phyllotaxis]]) of plants were frequently expressed as fractions involving Fibonacci numbers.<ref>{{Citation |first =Franck |last = Varenne |title = Formaliser le vivant - Lois, Théories, Modèles | accessdateaccess-date = 2022-10-30| url = https://www.numilog.com/LIVRES/ISBN/9782705670894.Livre | page = 28 | date = 2010| isbn = 9782705678128|publisher = Hermann|quote = En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].|langlanguage = fr}}</ref>

[[Przemysław Prusinkiewicz]] advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on [[free group]]s, specifically as certain [[L-system|Lindenmayer grammars]].<ref>{{Citation|first1 = Przemyslaw |last1 = Prusinkiewicz | first2 = James | last2 = Hanan| title = Lindenmayer Systems, Fractals, and Plants (Lecture Notes in Biomathematics) |publisher= [[Springer Science+Business Media|Springer-Verlag]] |year=1989 |isbn=978-0-387-97092-9}}</ref>

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<math display=block>\theta = \frac{2\pi}{\varphi^2} n,\ r = c \sqrt{n}</math>

where {{mvar|n}} is the index number of the floret and {{mvar|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 {{math|''F''(&thinsp;''j''):''F''(&thinsp;''j'' + 1)}}, the nearest neighbors of floret number {{mvar|n}} are those at {{math|''n'' ± ''F''(&thinsp;''j'')}} for some index {{mvar|j}}, which depends on {{mvar|r}}, the distance from the center. Sunflowers and similar flowers most commonly have spirals of florets in clockwise and counter-clockwise directions in the amount of adjacent Fibonacci numbers,{{sfn|Livio|2003|p=112}} typically counted by the outermost range of radii.<ref>{{Citation | last1 = Prusinkiewicz | first1 = Przemyslaw | author1-link = Przemyslaw Prusinkiewicz | author2-link = Aristid Lindenmayer | last2 = Lindenmayer | first2 = Aristid | title = The Algorithmic Beauty of Plants | publisher = Springer-Verlag | year = 1990 | pages = [https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] | chapter = 4 | chapter-url = https://algorithmicbotany.org/papers/#webdocs | isbn = 978-0-387-97297-8 | url = https://archive.org/details/algorithmicbeaut0000prus/page/101 }}</ref>

Fibonacci numbers also appear in the ancestral pedigrees of [[bee]]s (which are [[haplodiploid]]s), according to the following rules:

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* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}

* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.

* In [[software development]], Fibonacci numbers are often used by [[Agile management|agile]] teams operating under the [[Scrum (software development)|Scrum]] framework to size their [[product backlog]] items.<ref>{{cite web |last1=Kathuria |first1=Madhur |title=A Guide to Using the Fibonacci Sequence in Scrum |url=https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum |website=Scrum Alliance |access-date=8 August 2025}}</ref>

== See also ==

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| last1 =Borwein

| first1 =Jonathan M.

| authorlinkauthor-link =Jonathan Borwein

| authorlink2author-link2=Peter Borwein|first2=Peter B.|last2= Borwein

| title =Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

| pages =91–101

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* {{YouTube|id=hbUQlrLDAgw|title=Fibonacci Sequence and Golden Ratio: Mathematics in the Modern World - Mathuklasan with Sir Ram}} - animation of sequence, spiral, golden ratio, rabbit pair growth. Examples in art, music, architecture, nature, and astronomy

* [https://www.mathpages.com/home/kmath078/kmath078.htm Periods of Fibonacci Sequences Mod m] at MathPages

* [http://www.physorg.com/news97227410.html Scientists find clues to the formation of Fibonacci spirals in nature]