Editing Phyllotaxis

{{Short description|Arrangement of leaves on the stem of a plant}}
[[File:Aloe polyphylla 1.jpg|thumb|275px|Crisscrossing spirals of ''[[Aloe polyphylla]]'']]

In [[botany]], '''phyllotaxis''' ({{etymology|grc|''{{wikt-lang|grc|φύλλον}}'' ({{grc-transl|φύλλον}})|leaf||''{{wikt-lang|grc|τάξις}}'' ({{grc-transl|τάξις}})|arrangement}})<ref>{{LSJ|fu/llon|φύλλον}}, {{LSJ|ta/cis|τάξις|ref}}</ref> or '''phyllotaxy''' is the arrangement of [[leaf|leaves]] on a [[plant stem]]. Phyllotactic spirals form a distinctive class of [[patterns in nature]].

== Leaf arrangement ==
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The basic [[leaf#Arrangement on the stem|arrangements of leaves on a stem]] are '''opposite''' and '''alternate''' (also known as '''spiral'''). Leaves may also be '''[[Whorl (botany)|whorled]]''' if several leaves arise, or appear to arise, from the same level (at the same [[Node (botany)|node]]) on a stem.

[[File:Veronicastrum virg album C.jpg|thumb|upright|''[[Veronicastrum virginicum]]'' has whorls of leaves separated by long [[Plant stem|internodes]].]]

With an opposite leaf arrangement, two leaves arise from the stem at the same level (at the same [[Node (botany)|node]]), on opposite sides of the stem. An opposite leaf pair can be thought of as a whorl of two leaves.

With an alternate (spiral) pattern, each leaf arises at a different point (node) on the stem.

[[File:Clivia-GreenPlants.ca.jpg|thumb|right|Distichous leaf arrangement in ''[[Clivia]]'']]

'''Distichous''' phyllotaxis, also called "two-ranked leaf arrangement" is a special case of either opposite or alternate leaf arrangement where the leaves on a stem are arranged in two vertical columns on opposite sides of the stem. Examples include various [[bulb|bulbous plants]] such as ''[[Boophone]]''. It also occurs in other plant [[Habit (biology)|habits]] such as those of ''[[Gasteria]]'' or ''[[Aloe]]'' seedlings, and also in mature plants of related species such as ''[[Kumara plicatilis]]''.
[[File:Lithops 2.jpg|thumb|A ''[[Lithops]]'' species, showing its decussate growth in which a single pair of leaves is replaced at a time, leaving just one live active pair of leaves as the old pair withers]]
In an opposite pattern, if successive leaf pairs are 90 degrees apart, this habit is called '''[[decussate]]'''. It is common in members of the family [[Crassulaceae]]<ref name="Eggli2012">{{cite book| first = Urs | last = Eggli | name-list-style = vanc |title=Illustrated Handbook of Succulent Plants: Crassulaceae|url=https://books.google.com/books?id=nU7mCAAAQBAJ&pg=PA40|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-3-642-55874-0|pages=40–}}</ref> Decussate phyllotaxis also occurs in the [[Aizoaceae]]. In genera of the Aizoaceae, such as ''[[Lithops]]'' and ''[[Conophytum]]'', many species have just two fully developed leaves at a time, the older pair folding back and dying off to make room for the decussately oriented new pair as the plant grows.<ref name="Hartmann2012">{{cite book| first = Heidrun E.K. | last = Hartmann | name-list-style = vanc |title=Illustrated Handbook of Succulent Plants: Aizoaceae A–E|url=https://books.google.com/books?id=7oHuCAAAQBAJ&pg=PA14|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-3-642-56306-5|pages=14–}}</ref>

If the arrangement is both distichous and decussate, it is called '''secondarily distichous'''.

[[File:Ulistnienie (Phyllotaxys opposite-decussate).png|thumb|upright|A [[Decussation|decussate]] leaf pattern]]
[[File:Crassula rupestris-PICT3087.jpg|thumb|[[Decussation|Decussate]] phyllotaxis of ''[[Crassula]] rupestris'']]

The whorled arrangement is fairly unusual on plants except for those with particularly short [[internode (botany)|internodes]]. Examples of trees with whorled phyllotaxis are ''[[Brabejum stellatifolium]]''<ref name= "Marloth4">{{cite book | last = Marloth | first = Rudolf | name-list-style = vanc | title = The Flora of South Africa | date = 1932 | location = Cape Town & London | publisher = Darter Bros., Wheldon & Wesley }}</ref> and the related genus ''[[Macadamia]]''.<ref name= "RHSDict">{{cite book | last = Chittenden | first = Fred J. | name-list-style = vanc | publisher = Royal Horticultural Society | title = Dictionary of Gardening | location = Oxford | date = 1951 }}</ref>

A whorl can occur as a [[anatomical terms of location|basal]] structure where all the leaves are attached at the base of the shoot and the internodes are small or nonexistent. A basal whorl with a large number of leaves spread out in a circle is called a [[Rosette (botany)|rosette]].

== Repeating spiral ==
The rotational angle from leaf to leaf in a repeating spiral can be represented by a fraction of a [[full rotation]] around the stem.

Alternate distichous leaves will have an angle of 1/2 of a full rotation. In [[beech]] and [[hazel]] the angle is 1/3,{{Citation needed|date=December 2022|reason = in hazel (Corylus avellana) this is only true for vertical twigs; horizontal twigs are alternate distichous. Need a more detailed (and perhaps more botanical) source that covers this.}} in [[oak]] and [[apricot]] it is 2/5, in [[sunflowers]], [[Populus|poplar]], and [[pear]], it is 3/8, and in [[willow]] and [[almond]] the angle is 5/13.<ref>{{Cite book|title=Introduction to geometry|first=H. S. M. |last=Coxeter| name-list-style = vanc |author-link =Harold Scott MacDonald Coxeter|publisher =Wiley|year=1961|pages=169}}</ref> The numerator and denominator normally consist of a [[Fibonacci number]] and its second successor. The number of leaves is sometimes called rank, in the case of simple Fibonacci ratios, because the leaves line up in vertical rows. With larger Fibonacci pairs, the pattern becomes complex and non-repeating. This tends to occur with a basal configuration. Examples can be found in [[Asteraceae|composite]] [[flower]]s and [[seed]] heads. The most famous example is the [[sunflower]] head. This phyllotactic pattern creates an optical effect of criss-crossing spirals. In the botanical literature, these designs are described by the number of counter-clockwise spirals and the number of clockwise spirals. These also turn out to be [[Fibonacci numbers]]. In some cases, the numbers appear to be multiples of Fibonacci numbers because the spirals consist of whorls.

==Determination==
The pattern of leaves on a plant is ultimately controlled by the accumulation of the plant hormone [[auxin]] in certain areas of the [[meristem]].<ref>{{Cite journal |last1=Reinhardt |first1=Didier |last2=Mandel |first2=Therese |last3=Kuhlemeier |first3=Cris |date=April 2000 |title=Auxin Regulates the Initiation and Radial Position of Plant Lateral Organs |journal=The Plant Cell |language=en |volume=12 |issue=4 |pages=507–518 |doi=10.1105/tpc.12.4.507 |issn=1040-4651 |pmc=139849 |pmid=10760240|bibcode=2000PlanC..12..507R }}</ref><ref>{{cite journal | vauthors = Traas J, Vernoux T | title = The shoot apical meristem: the dynamics of a stable structure | journal = Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences | volume = 357 | issue = 1422 | pages = 737–47 | date = June 2002 | pmid = 12079669 | pmc = 1692983 | doi = 10.1098/rstb.2002.1091 }}</ref> Leaves become initiated in localized areas where auxin concentration is higher.{{Disputed inline|More mechanisms please|date=May 2016}} When a leaf is initiated and begins development, auxin begins to flow towards it, thus depleting auxin from area on the [[meristem]] close to where the leaf was initiated. This gives rise to a self-propagating system that is ultimately controlled by the ebb and flow of auxin in different regions of the [[meristem]]atic [[topography]].<ref>{{Cite journal |last1=Deb |first1=Yamini |last2=Marti |first2=Dominik |last3=Frenz |first3=Martin |last4=Kuhlemeier |first4=Cris |last5=Reinhardt |first5=Didier |date=2015-06-01 |title=Phyllotaxis involves auxin drainage through leaf primordia |journal=Development |language=en |volume=142 |issue=11 |pages=1992–2001 |doi=10.1242/dev.121244 |pmid=25953346 |s2cid=13800404 |issn=1477-9129|doi-access=free }}</ref><ref>{{cite journal | vauthors = Smith RS | title = The role of auxin transport in plant patterning mechanisms | journal = PLOS Biology | volume = 6 | issue = 12 | pages = e323 | date = December 2008 | pmid = 19090623 | pmc = 2602727 | doi = 10.1371/journal.pbio.0060323 | doi-access = free }}</ref>

== History ==
{{see also|Patterns in nature#History}}
Some early scientists—notably [[Leonardo da Vinci]]—made observations of the spiral arrangements of plants.<ref>{{cite book |author=Leonardo da Vinci |author-link=Leonardo da Vinci |editor-last=Taylor |editor-first=Pamela |title=The Notebooks of Leonardo da Vinci |publisher=New American Library|year=1971|page=121}}</ref> In 1754, [[Charles Bonnet]] observed that the spiral phyllotaxis of plants were frequently expressed in both [[clockwise]] and counter-clockwise [[golden ratio]] series.<ref name=livio110>{{cite book|last=Livio|first=Mario|author-link=Mario Livio|title=The Golden Ratio: The Story of Phi, the World's Most Astonishing Number|url=https://books.google.com/books?id=bUARfgWRH14C|orig-year=2002|edition=First trade paperback|year=2003|publisher=[[Random House|Broadway Books]]|location=New York City|isbn=978-0-7679-0816-0|page=110}}</ref> Mathematical observations of phyllotaxis followed with [[Karl Friedrich Schimper]] and his friend [[Alexander Braun]]'s 1830 and 1830 work, respectively; [[Auguste Bravais]] and his brother Louis connected phyllotaxis ratios to the [[Fibonacci sequence]] in 1837.<ref name=livio110/>

Insight into the mechanism had to wait until [[Wilhelm Hofmeister]] proposed a model in 1868. A [[primordium]], the nascent leaf, forms at the least crowded part of the shoot [[meristem]]. The [[golden angle]] between successive leaves is the blind result of this jostling. Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge. The generative spiral is a consequence of the same process that produces the clockwise and counter-clockwise spirals that emerge in densely packed plant structures, such as ''[[Protea]]'' flower disks or pinecone scales.

In modern times, researchers such as [[Mary Snow]] and George Snow<ref>{{cite journal |author1=Snow, M. |author2=Snow, R. |year=1934 |title=The interpretation of Phyllotaxis |journal=Biological Reviews |volume=9 |issue=1 |pages=132–137 |doi=10.1111/j.1469-185X.1934.tb00876.x|s2cid=86184933 }}</ref> continued these lines of inquiry. Computer modeling and morphological studies have confirmed and refined Hoffmeister's ideas. Questions remain about the details. Botanists are divided on whether the control of leaf migration depends on chemical [[gradient]]s among the [[Primordium|primordia]] or purely mechanical forces. [[Lucas number]]s rather than Fibonacci numbers have been observed in a few plants<ref>{{Cite book |last=Church |first=Arthur Harry |url=https://books.google.com/books?id=9cdJAQAAMAAJ&dq=Church,+A.H.+(1904).+On+the+Relation+of+phyllotaxis+to+mechanical+laws.+Williams+and+Norgate,+London.&pg=PA109 |title=On the Relation of Phyllotaxis to Mechanical Laws |date=1904 |publisher=Williams & Norgate |pages=198 |language=en}}</ref> and occasionally, the leaf positioning appears to be random.{{citation needed|date=March 2023}}

==Mathematics==
[[File:phyllotaxis_golden_angle.svg|thumb|End-on view of a plant stem showing consecutive leaves separated by the [[golden angle]] ]]
Physical models of phyllotaxis date back to [[George Biddell Airy|Airy]]'s experiment of packing hard spheres. [[Gerrit van Iterson]] diagrammed grids imagined on a cylinder (rhombic lattices).<ref>{{cite web | url=http://www.math.smith.edu/phyllo/About/History/VanIterson01.html | title=History | publisher=Smith College | access-date=24 September 2013 | archive-url=https://web.archive.org/web/20130927224034/http://www.math.smith.edu/phyllo/About/History/VanIterson01.html | archive-date=27 September 2013 | url-status=dead }}</ref> Douady et al. showed that phyllotactic patterns emerge as self-organizing processes in dynamic systems.<ref>{{cite journal | vauthors = Douady S, Couder Y | title = Phyllotaxis as a physical self-organized growth process | journal = Physical Review Letters | volume = 68 | issue = 13 | pages = 2098–2101 | date = March 1992 | pmid = 10045303 | doi = 10.1103/PhysRevLett.68.2098 | bibcode = 1992PhRvL..68.2098D }}</ref> In 1991, Levitov proposed that lowest energy configurations of repulsive particles in cylindrical geometries reproduce the spirals of botanical phyllotaxis.<ref>{{cite journal |author=Levitov LS |title=Energetic Approach to Phyllotaxis |journal=Europhys. Lett. |volume=14 |issue=6 |pages=533–9 |date=15 March 1991 |doi= 10.1209/0295-5075/14/6/006 |bibcode = 1991EL.....14..533L |s2cid=250864634 }}<br/>
{{cite journal | vauthors = Levitov LS | title = Phyllotaxis of flux lattices in layered superconductors | journal = Physical Review Letters | volume = 66 | issue = 2 | pages = 224–227 | date = January 1991 | pmid = 10043542 | doi = 10.1103/PhysRevLett.66.224 | bibcode = 1991PhRvL..66..224L }}</ref> More recently, Nisoli et al. (2009) showed that to be true by constructing a "magnetic cactus" made of magnetic dipoles mounted on bearings stacked along a "stem".<ref>{{cite journal | vauthors = Nisoli C, Gabor NM, Lammert PE, Maynard JD, Crespi VH | title = Static and dynamical phyllotaxis in a magnetic cactus | journal = Physical Review Letters | volume = 102 | issue = 18 | pages = 186103 | date = May 2009 | pmid = 19518890 | doi = 10.1103/PhysRevLett.102.186103 | arxiv = cond-mat/0702335 | bibcode = 2009PhRvL.102r6103N | s2cid = 4596630 }}</ref><ref>{{cite journal | vauthors = Nisoli C | title = Spiraling solitons: A continuum model for dynamical phyllotaxis of physical systems | journal = Physical Review E | volume = 80 | issue = 2 Pt 2 | pages = 026110 | date = August 2009 | pmid = 19792203 | doi = 10.1103/PhysRevE.80.026110 | url = https://www.researchgate.net/publication/26858941 | arxiv = 0907.2576 | bibcode = 2009PhRvE..80b6110N | s2cid = 27552596 }}</ref> They demonstrated that these interacting particles can access novel dynamical phenomena beyond what botany yields: a "dynamical phyllotaxis" family of non local topological [[solitons]] emerge in the [[nonlinear]] regime of these systems, as well as purely classical [[roton]]s and [[Maxon excitation|maxons]] in the spectrum of linear excitations.

Close packing of spheres generates a dodecahedral tessellation with pentaprismic faces. Pentaprismic symmetry is related to the Fibonacci series and the [[golden section]] of classical geometry.<ref name="Matila Ghyka 1946">{{Cite book | title = The Geometry of Art and Life | first = Matila | last = Ghyka | name-list-style = vanc | publisher = Dover | isbn = 978-0-486-23542-4 | url = https://archive.org/details/geometryofartlif00mati | year = 1977 }}</ref><ref>{{cite book|last=Adler|first=Irving|name-list-style=vanc|title=Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and the Golden Ratio Occur On Plants|title-link= Solving the Riddle of Phyllotaxis}}</ref>

==In art and architecture==

Phyllotaxis has been used as an inspiration for a number of sculptures and architectural designs. Akio Hizume has built and exhibited several bamboo towers based on the Fibonacci sequence which exhibit phyllotaxis.<ref>{{cite web| title=Star Cage | url=http://www.starcage.org/englishindex.html | author=Akio Hizume | access-date=18 November 2012 }}</ref> Saleh Masoumi has proposed a design for an apartment building in which the apartment [[balcony|balconies]] project in a spiral arrangement around a central axis and none shade the balcony of the apartment directly beneath.<ref>{{cite journal|journal=World Architecture News.com|date=11 Dec 2012|title=Open to the elements|url=http://www.worldarchitecturenews.com/index.php?fuseaction=wanappln.projectview&upload_id=21413}}</ref>

== See also ==
{{commons category|Phyllotaxis|lcfirst=yes}}

* [[Anisophylly]], a type of leaf size difference on horizontal shoots
* [[Available space theory]]
* [[Decussation]]
* [[Fermat's spiral]]
* [[L-system]]
* ''[[Orixa japonica]]''
* [[Parastichy]]
* [[Plastochron]]
* [[Repulsion theory]]
* [[Three-gap theorem]]
* [[Sphere packing in a cylinder]]
{{clear}}

== References ==
{{reflist|32em}}

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{{Patterns in nature}}

[[Category:Plant morphology]]
[[Category:Leaves]]