Editing Fibonacci sequence (section)

=== Nature ===
{{Further|Patterns in nature}}
{{see also|Golden ratio#Nature}}
[[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 | accessdate = 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 [...].|lang = 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>

[[File:SunflowerModel.svg|thumb|Illustration of Vogel's model for {{math|''n'' {{=}} 1 ... 500}}]]
A model for the pattern of [[floret]]s in the head of a [[sunflower]] was proposed by {{ill|Helmut Vogel|de|Helmut Vogel (Physiker)}} in 1979.<ref>{{Citation | last =Vogel | first =Helmut | title =A better way to construct the sunflower head | journal = Mathematical Biosciences | issue =3–4 | pages = 179–89 | year = 1979 | doi = 10.1016/0025-5564(79)90080-4 | volume = 44}}</ref>  This has the form

<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:
* If an egg is laid but not fertilized, it produces a male (or [[Drone (bee)|drone bee]] in honeybees).
* If, however, an egg is fertilized, it produces a female.

Thus, a male bee always has one parent, and a female bee has two. If one traces the pedigree of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. The number of ancestors at each level, {{math|''F''<sub>''n''</sub>}}, is the number of female ancestors, which is {{math|''F''<sub>''n''−1</sub>}}, plus the number of male ancestors, which is {{math|''F''<sub>''n''−2</sub>}}.<ref>{{Citation | url = https://www.fq.math.ca/Scanned/1-1/basin.pdf | title = The Fibonacci sequence as it appears in nature | journal = The Fibonacci Quarterly | volume = 1 | number = 1 | pages = 53–56 | year = 1963| doi = 10.1080/00150517.1963.12431602 | last1 = Basin | first1 = S. L. }}</ref><ref>Yanega, D. 1996. Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae). J. Kans. Ent. Soc. 69 Suppl.: 98-115.</ref> This is under the unrealistic assumption that the ancestors at each level are otherwise unrelated.

[[File:X chromosome ancestral line Fibonacci sequence.svg|thumb|upright=1.2|The number of possible ancestors on the X chromosome inheritance line at a given ancestral generation follows the Fibonacci sequence. (After Hutchison, L. "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships".<ref name="xcs"/>)]]

It has similarly been noticed that the number of possible ancestors on the human [[X chromosome]] inheritance line at a given ancestral generation also follows the Fibonacci sequence.<ref name="xcs">{{citation|last=Hutchison|first=Luke|date=September 2004|title=Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships|url=http://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|journal=Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04)|access-date=2016-09-03|archive-date=2020-09-25|archive-url=https://web.archive.org/web/20200925132536/https://fhtw.byu.edu/static/conf/2005/hutchison-growing-fhtw2005.pdf|url-status=dead}}</ref> A male individual has an X chromosome, which he received from his mother, and a [[Y chromosome]], which he received from his father. The male counts as the "origin" of his own X chromosome (<math>F_1=1</math>), and at his parents' generation, his X chromosome came from a single parent {{nowrap|(<math>F_2=1</math>)}}. The male's mother received one X chromosome from her mother (the son's maternal grandmother), and one from her father (the son's maternal grandfather), so two grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_3=2</math>)}}. The maternal grandfather received his X chromosome from his mother, and the maternal grandmother received X chromosomes from both of her parents, so three great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_4=3</math>)}}. Five great-great-grandparents contributed to the male descendant's X chromosome {{nowrap|(<math>F_5=5</math>)}}, etc. (This assumes that all ancestors of a given descendant are independent, but if any genealogy is traced far enough back in time, ancestors begin to appear on multiple lines of the genealogy, until eventually a [[Founder effect|population founder]] appears on all lines of the genealogy.)
[[File:BerlinVictoryColumnStairs.jpg|thumb|The Fibonacci sequence can also be found in man-made construction, as seen when looking at the staircase inside the Berlin Victory Column.]]