View source for Patterns in nature

{{short description|Visible regularity of form found in the natural world}}
{{good article}}
{{Use British English|date=April 2020}}
[[File:The Desert Ultra - Sand Dunes.jpg|thumb|Natural patterns form as wind blows sand in the dunes of the [[Namib Desert]]. The [[barchan|crescent shaped dunes]] and the [[capillary wave|ripples]] on their surfaces repeat wherever there are suitable conditions.]]
[[File:Yemen Chameleon (cropped).jpg|thumb|Patterns of the [[veiled chameleon]], ''Chamaeleo calyptratus'', provide [[camouflage]] and signal [[aposematism|mood]] as well as [[sexual selection|breeding condition]].]]

'''Patterns in nature''' are visible regularities of form found in the [[nature|natural world]]. These [[patterns]] recur in different contexts and can sometimes be [[modelled mathematically]]. Natural patterns include [[symmetries]], [[trees]], [[spirals]], [[meanders]], [[wave]]s, [[foams]], [[tessellations]], [[fracture|cracks]] and stripes.{{sfn|Stevens|1974|page=3}} Early [[Greek philosophers]] studied pattern, with [[Plato]], [[Pythagoras]] and [[Empedocles]] attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

In the 19th century, the Belgian physicist [[Joseph Plateau]] examined [[soap film]]s, leading him to formulate the concept of a [[minimal surface]]. The German biologist and artist [[Ernst Haeckel]] painted hundreds of [[Marine biology|marine organisms]] to emphasise their [[symmetry]]. Scottish biologist [[D'Arcy Thompson]] pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician [[Alan Turing]] predicted mechanisms of [[morphogenesis]] which give rise to [[Turing patterns|patterns]] of spots and stripes. The Hungarian biologist [[Aristid Lindenmayer]] and the French American mathematician [[Benoît Mandelbrot]] showed how the mathematics of [[fractals]] could create plant growth patterns.

[[Mathematics]], [[physics]] and [[chemistry]] can explain patterns in nature at different levels and scales. Patterns in living things are explained by the [[biology|biological]] processes of [[natural selection]] and [[sexual selection]]. Studies of [[pattern formation]] make use of [[computer models]] to simulate a wide range of patterns.

==History==

Early Greek philosophers attempted to explain order in [[nature]], anticipating modern concepts. [[Pythagoras]] (c.&nbsp;570–c.&nbsp;495&nbsp;BC) explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.{{efn|The so-called [[Pythagoreanism|Pythagoreans]], who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things. [[Aristotle]], ''Metaphysics 1–5 '', c. 350 BC}} [[Empedocles]] (c.&nbsp;494–c.&nbsp;434&nbsp;BC) to an extent anticipated [[Charles Darwin|Darwin]]'s evolutionary explanation for the structures of organisms.{{efn|Aristotle reports Empedocles arguing that, "[w]herever, then, everything turned out as it would have if it were happening for a purpose, there the creatures survived, being accidentally compounded in a suitable way; but where this did not happen, the creatures perished." ''The Physics'', B8, 198b29 in Kirk, et al., 304).}} [[Plato]] (c.&nbsp;427–c.&nbsp;347&nbsp;BC) argued for the existence of natural [[Universals (metaphysics)|universals]]. He considered these to consist of [[Theory of Forms|ideal forms]] ({{lang|grc|εἶδος}} ''eidos'': "form") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect circle.<ref>{{cite web |url=http://plato.stanford.edu/entries/platonism/ |website=[[Stanford Encyclopedia of Philosophy]] |title=Platonism in Metaphysics |orig-year=2004 |date=7 April 2009 |access-date=4 May 2012 |author=Balaguer, Mark}}</ref> [[Theophrastus]] (c.&nbsp;372–c.&nbsp;287&nbsp;BC) noted that plants "that have flat leaves have them in a regular series"; [[Pliny the Elder]] (23–79&nbsp;AD) noted their patterned circular arrangement.<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 |isbn=978-0-7679-0816-0 |page=110}}</ref> Centuries later, [[Leonardo da Vinci]] (1452–1519) noted the spiral arrangement of leaf patterns, that tree trunks gain successive rings as they age, and proposed [[#Da Vinci branching rule|a rule]] purportedly satisfied by the cross-sectional areas of tree-branches.<ref>{{cite book |last=Da Vinci |first=Leonardo |editor-last=Taylor |editor-first=Pamela |title=The Notebooks of Leonardo da Vinci |publisher=New American Library |year=1971 |page=121}}</ref><ref name=livio110/>

In 1202, [[Fibonacci|Leonardo Fibonacci]] introduced the [[Fibonacci sequence]] to the western world with his book {{Lang|la|[[Liber Abaci]]}}.<ref>{{cite journal |last=Singh |first=Parmanand |title=Acharya Hemachandra and the (so called) Fibonacci Numbers |journal=Mathematics Education Siwan |year=1986 |volume=20 |issue=1 |pages=28–30 |issn=0047-6269 }}</ref> Fibonacci presented a [[thought experiment]] on the growth of an idealized [[rabbit]] population.<ref>{{cite web |last=Knott |first=Ron |title=Fibonacci's Rabbits |url=http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits |website=[[University of Surrey]] Faculty of Engineering and Physical Sciences}}</ref> [[Johannes Kepler]] (1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the [[pentagon]]al form of some flowers.<ref name=livio110/> In 1658, the English physician and philosopher [[Sir Thomas Browne]] discussed "how Nature Geometrizeth" in ''[[The Garden of Cyrus]]'', citing [[Pythagorean numerology]] involving the number 5, and the [[Platonic form]] of the [[quincunx]] pattern. The discourse's central chapter features examples and observations of the quincunx in botany.<ref>{{cite book |author=Browne, Thomas |author-link=Sir Thomas Browne |contribution-url=http://penelope.uchicago.edu/gardennoframes/garden3.html |chapter=Chapter III |title=[[The Garden of Cyrus]] |date=1658}}</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/> 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, also noting its appearance in [[pinecone]]s and [[pineapple]]s.<ref name=livio110/> In his 1854 book, German psychologist [[Adolf Zeising]] explored the golden ratio expressed in the arrangement of plant parts, the [[skeleton]]s of animals and the branching patterns of their veins and nerves, as well as in [[crystal]]s.<ref>{{Cite book |title=Proportion: Science, Philosophy, Architecture |author=Padovan, Richard |author-link=Richard Padovan |publisher=Taylor & Francis |date=1999 |isbn=978-0-419-22780-9 |pages=305–306 |url=https://books.google.com/books?id=Vk_CQULdAssC&pg=PA306}}</ref><ref>{{cite journal |journal=Nexus Network Journal |first=Richard |last=Padovan |author-link=Richard Padovan |title=Proportion: Science, Philosophy, Architecture |volume=4 |pages=113–122 |doi=10.1007/s00004-001-0008-7 |date=2002 |issue=1 |doi-access=free}}</ref><ref>{{cite book |first=Adolf |last=Zeising |title=Neue Lehre van den Proportionen des meschlischen Körpers |date=1854 |page=preface |no-pp=true}}</ref>

In the 19th century, the Belgian physicist [[Joseph Plateau]] (1801–1883) formulated the [[Plateau's problem|mathematical problem]] of the existence of a [[minimal surface]] with a given boundary, which is now named after him. He studied soap films intensively, formulating [[Plateau's laws]] which describe the structures formed by films in foams.{{sfn|Stewart|2001|pages=108–109}} [[Lord Kelvin]] identified the problem of the most efficient way to pack cells of equal volume as a foam in 1887; his solution uses just one solid, the [[bitruncated cubic honeycomb]] with very slightly curved faces to meet Plateau's laws. No better solution was found until 1993 when Denis Weaire and Robert Phelan proposed the [[Weaire–Phelan structure]]; the [[Beijing National Aquatics Center]] adapted the structure for their outer wall in the [[2008 Summer Olympics]].{{sfn|Ball|2009a|pages=73–76}} [[Ernst Haeckel]] (1834–1919) painted beautiful illustrations of marine organisms, in particular [[Radiolaria]], emphasising their [[symmetry]] to support his faux-[[Charles Darwin|Darwinian]] theories of evolution.{{sfn|Ball|2009a|page=41}} The American photographer [[Wilson Bentley]] took the first micrograph of a [[snowflake]] in 1885.<ref>{{Cite book |title=Encyclopedia of Nineteenth-Century Photography |volume=1 |first=John |last=Hannavy |publisher=CRC Press |date=2007 |isbn=978-0-415-97235-2 |page=149 |url=https://books.google.com/books?id=PJ8DHBay4_EC&pg=PA149}}</ref>

In the 20th century, [[Arthur Harry Church|A. H. Church]] studied the patterns of phyllotaxis in his 1904 book.<ref name=livio111>{{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 |year=2003|publisher=[[Random House|Broadway Books]]|location=New York |isbn=978-0-7679-0816-0|page=111}}</ref> In 1917, [[D'Arcy Wentworth Thompson]] published ''[[On Growth and Form]]''; his description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants showed that simple equations could describe the spiral growth patterns of [[animal horn]]s and [[mollusc shell]]s.<ref>[http://www.darcythompson.org/about.html About D'Arcy] {{Webarchive|url=https://web.archive.org/web/20170701210817/http://www.darcythompson.org/about.html |date=2017-07-01 }}. D' Arcy 150. [[University of Dundee]] and the [[University of St Andrews]]. Retrieved 16 October 2012.</ref> In 1952, the computer scientist [[Alan Turing]] (1912–1954) wrote ''[[The Chemical Basis of Morphogenesis]]'', an analysis of the mechanisms that would be needed to create patterns in living organisms, in the process called [[morphogenesis]].<ref name=Turing>{{Cite journal |last=Turing |first=A. M. |author-link=Alan Turing |title=The Chemical Basis of Morphogenesis |journal=[[Philosophical Transactions of the Royal Society B]] |volume=237 |pages=37–72 |date=1952 |doi=10.1098/rstb.1952.0012 |issue=641 |bibcode=1952RSPTB.237...37T |s2cid=937133 |doi-access= }}</ref> He predicted [[Chemical clock|oscillating]] [[chemical reaction]]s, in particular the [[Belousov–Zhabotinsky reaction]]. These activator-inhibitor mechanisms can, Turing suggested, generate patterns (dubbed "[[Turing pattern]]s") of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis.{{sfn|Ball|2009a|pages=163, 247–250}}
In 1968, the Hungarian theoretical biologist [[Aristid Lindenmayer]] (1925–1989) developed the [[L-system]], a [[formal grammar]] which can be used to model [[plant development|plant growth patterns]] in the style of [[fractal]]s.<ref name=L>[[Rozenberg, Grzegorz]]; Salomaa, Arto. ''The Mathematical Theory of L Systems''. [[Academic Press]], New York, 1980. {{isbn|0-12-597140-0}}</ref> L-systems have an [[alphabet]] of symbols that can be combined using [[Formal grammar|production rules]] to build larger strings of symbols, and a mechanism for translating the generated strings into geometric structures. In 1975, after centuries of slow development of the mathematics of patterns by [[Gottfried Leibniz]], [[Georg Cantor]], [[Helge von Koch]], [[Wacław Sierpiński]] and others, [[Benoît Mandelbrot]] wrote a famous paper, ''[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]]'', crystallising mathematical thought into the concept of the [[fractal]].<ref name=Mandelbrot/>

<gallery mode=packed>
File:Cycas circinalis male cone in Olomouc.jpg|[[Fibonacci number]] patterns occur widely in plants such as this queen sago, ''[[Cycas circinalis]]''.
File:National Aquatics Center Construction (cropped).jpg|Beijing's National Aquatics Center for the 2008 Olympic games has a [[Weaire–Phelan structure]].
File:Drcy.svg|D'Arcy Thompson pioneered the study of growth and form in his 1917 book.
</gallery>

==Causes==
[[File:Aphids and live young under Sycamore leaf.JPG|thumb|upright|Composite patterns: [[aphids]] and newly born young in arraylike clusters on [[sycamore]] leaf, divided into [[polygon]]s by [[vein (botany)|veins]], which are avoided by the young aphids]]

Living things like [[orchid]]s, [[hummingbird]]s, and the [[Structural coloration|peacock's tail]] have abstract designs with a beauty of form, pattern and colour that artists struggle to match.<ref name=ForbesGuardian>Forbes, Peter. ''All that useless beauty''. The Guardian. Review: Non-fiction. 11 February 2012.</ref> The beauty that people perceive in nature has causes at different levels, notably in the mathematics that governs what patterns can physically form, and among living things in the effects of natural selection, that govern how patterns evolve.{{sfn|Stevens|1974|page=222}}

[[Mathematics]] seeks to discover and explain abstract patterns or regularities of all kinds.<ref name=Steen>{{cite journal |last1=Steen |first1=L.A. |author-link=Lynn Steen |year=1988 |title=The Science of Patterns |url=http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx |journal=[[Science (journal)|Science]] |volume=240 |issue=4852 |pages=611–616 |doi=10.1126/science.240.4852.611 |pmid=17840903 |bibcode=1988Sci...240..611S |s2cid=4849363 |access-date=2012-05-02 |archive-date=2010-10-28 |archive-url=https://web.archive.org/web/20101028101034/http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx |url-status=dead }}</ref><ref name=Devlin>[[Keith Devlin|Devlin, Keith]]. ''Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe'' (Scientific American Paperback Library) 1996</ref>
Visual patterns in nature find explanations in [[chaos theory]], fractals, logarithmic spirals, topology and other mathematical patterns. For example, [[L-system]]s form convincing models of different patterns of tree growth.<ref name=L/>

The laws of [[physics]] apply the abstractions of mathematics to the real world, often as if it were [[perfection#Physics and chemistry|perfect]]. For example, a [[crystal]] is perfect when it has no structural defects such as dislocations and is fully symmetric. Exact mathematical perfection can only approximate real objects.<ref name=Tatarkiewicz>{{cite journal |last=Tatarkiewicz |first=Władysław |author-link=Władysław Tatarkiewicz |title=Perfection in the Sciences. II. Perfection in Physics and Chemistry |journal=Dialectics and Humanism |volume=7 |issue=2 (spring 1980) |page=139}}</ref> Visible patterns in nature are governed by [[physical law]]s; for example, [[meander]]s can be explained using [[fluid dynamics]].

In [[biology]], [[natural selection]] can cause the development of patterns in living things for several reasons, including [[camouflage]],<ref name=DarwinCh4>[[Charles Darwin|Darwin, Charles]]. ''On the Origin of Species''. 1859, chapter 4.</ref> [[sexual selection]],<ref name=DarwinCh4/> and different kinds of signalling, including [[mimicry]]<ref name=Wickler>{{cite book |last=Wickler |first=Wolfgang |author-link=Wolfgang Wickler |date=1968 |title=Mimicry in plants and animals |publisher=McGraw-Hill |location=New York |url=https://archive.org/details/mimicryinplantsa00wick |url-access=registration}}</ref> and [[cleaning symbiosis]].<ref name=PoulinGrutter>{{cite journal |last1=Poulin |first1=R. |last2=Grutter |first2=A.S. |date=1996 |jstor=1312929 |title=Cleaning symbioses: proximate and adaptive explanations |journal=BioScience |volume=46 |issue=7 |pages=512–517 |doi=10.2307/1312929|doi-access=free }}</ref> In plants, the shapes, colours, and patterns of [[insect pollination|insect-pollinated]] [[flowers]] like the [[Lilium|lily]] have evolved to attract insects such as [[European honey bee|bees]]. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as [[nectar guides]] that can be seen at a distance.<ref>{{cite web |last=Koning |first=Ross |url=http://plantphys.info/plants_human/pollenadapt.shtml |title=Plant Physiology Information Website |website=Pollination Adaptations |date=1994 |access-date=2 May 2012 }}</ref>

== Types of pattern ==

=== Symmetry ===

{{further|Symmetry in biology|Floral symmetry|Crystal symmetry}}

[[Symmetry]] is pervasive in living things. Animals mainly have bilateral or [[Reflection symmetry|mirror symmetry]], as do the leaves of plants and some flowers such as [[orchid]]s.{{sfn|Stewart|2001|pages=48–49}} Plants often have radial or [[rotational symmetry]], as do many flowers and some groups of animals such as [[sea anemone]]s. Fivefold symmetry is found in the [[echinoderms]], the group that includes [[starfish]], [[sea urchin]]s, and [[sea lilies]].{{sfn|Stewart|2001|pages=64–65}}

Among non-living things, [[snowflake]]s have striking [[dihedral symmetry|sixfold symmetry]]; each flake's structure forms a record of the varying conditions during its crystallization, with nearly the same pattern of growth on each of its six arms.{{sfn|Stewart|2001|page=52}} [[Crystal]]s in general have a variety of symmetries and [[crystal habit]]s; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike [[quasicrystals]]).{{sfn|Stewart|2001|pages=82–84}} Rotational symmetry is found at different scales among non-living things, including the crown-shaped [[Splash (fluid mechanics)|splash]] pattern formed when a drop falls into a pond,{{sfn|Stewart|2001|page=60}} and both the [[spheroidal]] shape and rings of a [[planet]] like [[Saturn]].{{sfn|Stewart|2001|page=71}}

Symmetry has a variety of causes. Radial symmetry suits organisms like sea anemones whose adults do not move: food and threats may arrive from any direction. But animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialised with a mouth and sense organs ([[cephalisation]]), and the body becomes bilaterally symmetric (though internal organs need not be).<ref>{{cite web |url=http://biocongroup.eu/DA/Calendario_files/Bilateria.pdf |title=Animal Diversity |edition=Third |website=Chapter 8: Acoelomate Bilateral Animals |date=2002 |access-date=25 October 2012 |last1=Hickman |first1=Cleveland P. |author2=Roberts, Larry S. |author3=Larson, Allan |pages=139 |archive-url=http://arquivo.pt/wayback/20160517212058/http://biocongroup.eu/DA/Calendario_files/Bilateria.pdf |archive-date=17 May 2016 |url-status=dead }}</ref> More puzzling is the reason for the fivefold (pentaradiate) symmetry of the echinoderms. Early echinoderms were bilaterally symmetrical, as their larvae still are. Sumrall and Wray argue that the loss of the old symmetry had both developmental and ecological causes.<ref>{{cite journal |title=Ontogeny in the fossil record: diversification of body plans and the evolution of "aberrant" symmetry in Paleozoic echinoderms |last1=Sumrall |first1=Colin D. |last2=Wray |first2=Gregory A. |journal=Paleobiology |date=January 2007 |volume=33 |issue=1 |pages=149–163 |jstor=4500143 |doi=10.1666/06053.1 |bibcode=2007Pbio...33..149S |s2cid=84195721 }} <!--http://www.psjournals.org/doi/abs/10.1666/06053.1--></ref> In the case of [[ice eggs]], the gentle churn of water, blown by a suitably stiff breeze makes concentric layers of ice form on a seed particle that then grows into a floating ball as it rolls through the freezing currents.<ref>{{cite web |title=Image of the Week – Goodness gracious, great balls of ice! |url=https://blogs.egu.eu/divisions/cr/2016/11/18/image-of-the-week-goodness-gracious-great-balls-of-ice/ |access-date=2022-04-23 |website=Cryospheric Sciences}}</ref>

<gallery mode=packed class="center">
File:Tiger-berlin-5 symmetry.jpg|Animals often show mirror or [[bilateral symmetry]], like this [[tiger]].
File:Starfish 02 (paulshaffner) cropped.jpg|[[Echinoderms]] like this [[starfish]] have [[pentaradial symmetry|fivefold symmetry]].
File:Medlar 5-symmetry.jpg|Fivefold symmetry can be seen in many flowers and some fruits like this [[Mespilus germanica|medlar]].
File:Schnee2.jpg|[[Snowflake]]s have [[dihedral symmetry|sixfold symmetry]].
File:Aragonite-Fluorite-cflu02c.jpg|[[Fluorite]] showing cubic [[crystal habit]].
File:Water splashes 001.jpg|Water [[Splash (fluid mechanics)|splash]] approximates [[radial symmetry]].
File:GarnetCrystalUSGOV.jpg|[[Garnet]] showing rhombic dodecahedral crystal habit.
File:Two Oceans Aquarium03.jpg|[[Sea anemone]]s have [[rotational symmetry]].
File:Mikrofoto.de-volvox-8.jpg|''[[Volvox]]'' has spherical symmetry.
File:Jää on kulmunud pallideks (Looduse veidrused). 05.jpg|[[Ice eggs]] gain spherical symmetry by being rolled about by wind and currents.
</gallery>

===Trees, fractals ===

{{anchor|Da Vinci branching rule}}
The branching pattern of trees was described in the [[Italian Renaissance]] by [[Leonardo da Vinci]]. In ''[[A Treatise on Painting]]'' he stated that:

<blockquote>All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk [below them].<ref name=JPR>{{cite book |editor-last=Richter |editor-first=Jean Paul |orig-year=1880 |title=The Notebooks of Leonardo da Vinci |location=Dover |year=1970 |isbn=978-0-486-22572-2 |url=http://www.fromoldbooks.org/Richter-NotebooksOfLeonardo }}</ref></blockquote>

A more general version states that when a parent branch splits into two or more child branches, the surface areas of the child branches add up to that of the parent branch.<ref name=NPR>{{cite news |first=Joe |last=Palca |date=December 26, 2011 |title=The Wisdom of Trees (Leonardo Da Vinci Knew It) |work=Morning Edition |publisher=NPR |url=https://www.npr.org/2011/12/26/144127874/the-wisdom-of-trees-leonardo-da-vinci-knew-it |access-date=16 July 2019 }}</ref> An equivalent formulation is that if a parent branch splits into two child branches, then the cross-sectional diameters of the parent and the two child branches form a [[Pythagorean triplet|right-angled triangle]]. One explanation is that this allows trees to better withstand high winds.<ref name=NPR/> Simulations of biomechanical models agree with the rule.<ref name=plos>{{cite magazine |last1=Minamino |first1=Ryoko |last2=Tateno |first2=Masaki  |year=2014 |title=Tree Branching: Leonardo da Vinci's Rule versus Biomechanical Models |magazine=PLoS One |volume=9 |issue=4 |page=e93535 |doi=10.1371/journal.pone.0093535 |doi-access=free }}</ref>

[[Fractal]]s are infinitely [[self-similarity|self-similar]], iterated mathematical constructs having [[fractal dimension]].<ref name="Mandelbrot">{{cite book |last=Mandelbrot |first=Benoît B. |title=The fractal geometry of nature|date=1983 |publisher=Macmillan}}</ref><ref name="Falconer">{{Cite book |last=Falconer |first=Kenneth |author-link=Kenneth Falconer (mathematician) |title=Fractal Geometry: Mathematical Foundations and Applications |publisher=John Wiley |date=2003}}</ref><ref name="patterns">{{Cite book |title=Fractals:The Patterns of Chaos |last=Briggs |first=John |date=1992 |publisher=Thames and Hudson |page=148 }}</ref> Infinite [[iteration]] is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of [[fern]]s and [[umbellifer]]s (Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including [[bryozoa]], [[coral]]s, [[hydrozoa]] like the [[air fern]], ''Sertularia argentea'', and in non-living things, notably [[electrical discharge]]s. [[L-system|Lindenmayer system]] fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points ([[Internode (botany)|internode]] length), and number of branches per branch point.<ref name=L/>

Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, [[river|river networks]], geologic [[fault line]]s, [[mountain]]s, [[coastline]]s,<ref name=Batty>{{cite journal |last=Batty |first=Michael |title=Fractals – Geometry Between Dimensions |journal=New Scientist |date=4 April 1985 |page=31 |volume=105 |issue=1450}}</ref> [[animal coloration]], [[snowflake|snow flakes]],<ref name="snowflake">{{cite book |url=https://books.google.com/books?id=aHux78oQbbkC&q=snowflake+fractals+book&pg=PA25 |page=25 |author=Meyer, Yves |author2=Roques, Sylvie |title=Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications," Toulouse, France – June 1992 |date=1993 |publisher=Atlantica Séguier Frontières|isbn=9782863321300 }}</ref> [[crystal]]s,<ref name="crystal">{{cite book |last1=Carbone |first1=Alessandra |last2=Gromov |first2=Mikhael |last3=Prusinkiewicz |first3=Przemyslaw |title=Pattern formation in biology, vision and dynamics |date=2000 |publisher=World Scientific |isbn=978-9810237929 |page=78 }}</ref> [[blood vessel]] branching,<ref>{{cite book |chapter=Fractal aspects of three-dimensional vascular constructive optimization |last1=Hahn |first1=Horst K. |last2=Georg |first2=Manfred |last3=Peitgen |first3=Heinz-Otto |editor1=Losa, Gabriele A. |editor2=Nonnenmacher, Theo F. |title=Fractals in biology and medicine |date=2005 |publisher=Springer |pages=55–66}}</ref> [[Purkinje cell|Purkinje cells]],<ref>{{cite journal|last1=Takeda|first1=T|last2=Ishikawa|first2=A|last3=Ohtomo|first3=K|last4=Kobayashi|first4=Y|last5=Matsuoka|first5=T|title=Fractal dimension of dendritic tree of cerebellar Purkinje cell during onto- and phylogenetic development|journal=Neurosci Research|date=February 1992|volume=13|issue=1|pages=19–31|doi=10.1016/0168-0102(92)90031-7|pmid=1314350|s2cid=4158401}}</ref> [[actin cytoskeleton|actin cytoskeletons]],<ref>{{Cite journal |last=Sadegh |first=Sanaz |date=2017 |title=Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork |journal=Physical Review X |volume=7 |issue=1 |page=011031 |doi=10.1103/PhysRevX.7.011031 |pmc=5500227 |pmid=28690919 |arxiv=1702.03997 |bibcode=2017PhRvX...7a1031S}}</ref> and [[wind wave|ocean waves]].<ref name="nature">{{cite book |last=Addison |first=Paul S. |title=Fractals and chaos: an illustrated course |date=1997 |publisher=CRC Press |pages=44–46 }}</ref>

<gallery mode=packed class="center">
File:Dragon trees.jpg|The growth patterns of certain trees resemble these [[L-system|Lindenmayer system]] fractals.
File:Baobab Tree at Vasai Fort.jpg|Branching pattern of a [[baobab]] tree
File:Anthriscus sylvestris (Köhler's Medizinal-Pflanzen).jpg|Leaf of cow parsley, ''[[Anthriscus sylvestris]]'', is 2- or 3-[[pinnate]], not infinite
File:Romanesco broccoli (Brassica oleracea).jpg|[[Fractal]] spirals: [[Romanesco broccoli]] showing [[self-similar]] form
File:Angelica flowerhead showing pattern.JPG|[[Angelica]] flowerhead, a [[sphere]] made of spheres (self-similar)
File:Square1.jpg|Trees: [[Lichtenberg figure]]: high voltage [[dielectric]] breakdown in an [[Poly(methyl methacrylate)|acrylic polymer]] block
File:Dendritic Copper Crystals - 20x magnification.jpg|Trees: [[Dendrite (crystal)|dendritic]] copper crystals (in microscope)
<!--File:Fractal defrosting patterns on Mars.jpg|Defrosting patterns, polar Mars. Image c. 1&nbsp;km. wide-->
</gallery>

=== Spirals ===
{{further|Phyllotaxis}}

[[Spirals]] are common in plants and in some animals, notably [[molluscs]]. For example, in the [[nautilus]], a cephalopod mollusc, each [[Camera (cephalopod)|chamber]] of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a [[logarithmic spiral]].<ref>Maor, Eli. ''e: The Story of a Number''. Princeton University Press, 2009. Page 135.</ref> Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity.{{sfn|Ball|2009a|pages=29–32}}

Plant spirals can be seen in [[phyllotaxis]], the arrangement of leaves on a stem, and in the arrangement ([[parastichy]]<ref>{{cite web |url=http://www.math.smith.edu/phyllo/About/Lattices/SpiralLattices.html |archive-url=https://web.archive.org/web/20100526030701/http://www.math.smith.edu/phyllo/About/Lattices/SpiralLattices.html |url-status=dead |archive-date=26 May 2010 |title=Spiral Lattices & Parastichy |website=Smith College |access-date=24 September 2013 }}</ref>) of other parts as in [[compositae|composite]] [[flower|flower heads]] and [[seed|seed heads]] like the [[sunflower]] or [[fruit]] structures like the [[pineapple]]<ref name=livio111/><ref name=Kappraff>{{cite journal |url=http://www.scipress.org/journals/forma/pdf/1904/19040335.pdf |title=Growth in Plants: A Study in Number |author=Kappraff, Jay |journal=Forma |date=2004 |volume=19 |pages=335–354 |access-date=2012-05-02 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304001606/http://www.scipress.org/journals/forma/pdf/1904/19040335.pdf |url-status=dead }}</ref>{{rp|337}} and [[salak|snake fruit]], as well as in the pattern of scales in [[pine cones]], where multiple spirals run both clockwise and anticlockwise. These arrangements have explanations at different levels – mathematics, physics, chemistry, biology – each individually correct, but all necessary together.{{sfn|Ball|2009a|page=13}} Phyllotaxis spirals can be generated from [[Fibonacci number|Fibonacci ratios]]: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (each subsequent number being the sum of the two preceding ones). For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2. In [[hazel]] the ratio is 1/3; in [[apricot]] it is 2/5; in [[pear]] it is 3/8; in [[almond]] it is 5/13.<ref>{{Cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |title=Introduction to geometry |publisher=Wiley |date=1961 |pages=169 }}</ref> 

In disc phyllotaxis as in the [[sunflower]] and [[Bellis perennis|daisy]], the florets are arranged along [[Fermat's spiral]], but this is disguised because successive florets are spaced far apart, by the [[golden angle]], 137.508° (dividing the circle in the [[golden ratio]]); when the flowerhead is mature so all the elements are the same size, this spacing creates a Fibonacci number of more obvious spirals.<ref>{{cite book |last=Prusinkiewicz |first=Przemyslaw |author-link=Przemyslaw Prusinkiewicz |author2=Lindenmayer, Aristid |author-link2=Aristid Lindenmayer |title =The Algorithmic Beauty of Plants |publisher=Springer-Verlag |date=1990 |pages=[https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] |url=https://archive.org/details/algorithmicbeaut0000prus/page/101 |isbn=978-0-387-97297-8 }}</ref>

From the point of view of physics, spirals are lowest-energy configurations<ref>{{cite journal |last=Levitov |first=L. S. |title=Energetic Approach to Phyllotaxis |journal=Europhysics Letters |volume=14 |issue=6 |pages=533–539 |date=15 March 1991 |doi=10.1209/0295-5075/14/6/006 |bibcode=1991EL.....14..533L |s2cid=250864634 }}</ref> which emerge spontaneously through [[self-organizing]] processes in [[dynamic system]]s.<ref>{{cite journal |last1=Douady |first1=S. |last2=Couder |first2=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> From the point of view of chemistry, a spiral can be generated by a reaction-diffusion process, involving both activation and inhibition. Phyllotaxis is controlled by [[protein]]s that manipulate the concentration of the plant hormone [[auxin]], which activates [[meristem]] growth, alongside other mechanisms to control the relative angle of buds around the stem.{{sfn|Ball|2009a|pages=163, 249–250}} From a biological perspective, arranging leaves as far apart as possible in any given space is favoured by natural selection as it maximises access to resources, especially sunlight for [[photosynthesis]].<ref name=Kappraff/>

<gallery mode=packed class="center">
File:Fibonacci spiral 34.svg|[[Fibonacci number|Fibonacci]] spiral
File:Ovis canadensis 2 (cropped).jpg|[[Bighorn sheep]], ''Ovis canadensis''
File:Aloe polyphylla spiral.jpg|Spirals: [[phyllotaxis]] of spiral aloe, ''[[Aloe polyphylla]]''
File:NautilusCutawayLogarithmicSpiral.jpg|''[[Nautilus]]'' shell's [[logarithm]]ic growth spiral
File:Pflanze-Sonnenblume1-Asio (cropped).JPG|[[Fermat's spiral]]: seed head of [[sunflower]], ''Helianthus annuus''
File:Red Cabbage cross section showing spirals.jpg|Multiple Fibonacci spirals: red [[cabbage]] in cross section
File:Trochoidea liebetruti (Albers, 1852) (4308584755).jpg|Spiralling shell of ''[[Trochoidea liebetruti]]''
File:Fibonacci spin (cropped).jpg|Water droplets fly off a wet, spinning ball in [[equiangular spiral]]s
</gallery>

=== Chaos, flow, meanders ===

In mathematics, a [[dynamical system]] is chaotic if it is (highly) sensitive to initial conditions (the so-called "[[butterfly effect]]"<ref>{{cite journal |last=Lorenz |first=Edward N. |author-link=Edward Norton Lorenz |title=Deterministic Nonperiodic Flow|journal=Journal of the Atmospheric Sciences |date=March 1963 |volume=20|issue=2 |pages=130–141 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |bibcode=1963JAtS...20..130L |doi-access=free}}</ref>), which requires the mathematical properties of [[topological mixing]] and [[dense set|dense]] [[periodic orbits]].<ref>{{cite book |author=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |date=1999 |page=117 }}</ref>

Alongside fractals, [[chaos theory]] ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the ''[[strange attractors]]'' in chaotic systems have a [[fractal dimension]].<ref>{{cite book |author=Ruelle, David |title=Chance and Chaos |publisher=Princeton University Press |year=1991}}</ref> Some [[cellular automata]], simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably [[Stephen Wolfram]]'s [[Rule 30]].<ref>{{cite book |author=Wolfram, Stephen |title=[[A New Kind of Science]] |publisher=Wolfram Media |year=2002}}</ref>

[[Vortex street]]s are zigzagging patterns of whirling [[Vortex|vortices]] created by the unsteady [[flow separation|separation of flow]] of a [[fluid]], most often air or water, over obstructing objects.<ref>{{cite book |last=von Kármán |first=Theodore |title=Aerodynamics |publisher=McGraw-Hill |year=1963 |isbn=978-0070676022 }}. Dover (1994): {{isbn|978-0486434858}}.</ref> Smooth ([[Laminar flow|laminar]]) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the [[viscosity]] of the fluid.

[[Meander]]s are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as [[helical flow]] drags material like sand and gravel across the river to the inside of the bend. The outside of the loop is left clean and unprotected, so [[erosion]] accelerates, further increasing the meandering in a powerful [[positive feedback loop]].<ref>{{cite book |last=Lewalle |first=Jacques |title=Lecture Notes in Incompressible Fluid Dynamics: Phenomenology, Concepts and Analytical Tools |chapter=Flow Separation and Secondary Flow: Section 9.1 |chapter-url=http://www.ecs.syr.edu/faculty/lewalle/FluidDynamics/fluidsCh9.pdf |date=2006 |location=Syracuse, New York |publisher=Syracuse University |url-status=dead |archive-url=https://web.archive.org/web/20110929075022/http://www.ecs.syr.edu/faculty/lewalle/FluidDynamics/fluidsCh9.pdf |archive-date=29 September 2011 }}.</ref>

<gallery mode=packed class="center">
File:Textile cone (cropped).JPG|Chaos: shell of [[gastropod]] [[mollusc]] the cloth of gold cone, ''[[Conus textile]]'', resembles [[Rule 30]] [[cellular automaton]]
File:Vortex-street-1.jpg|Flow: [[vortex street]] of clouds at [[Juan Fernandez Islands]]
File:Rio Negro meanders.JPG|Meanders: dramatic [[meander scar]]s and [[oxbow lake]]s in the broad [[flood plain]] of the [[Río Negro (Argentina)|Rio Negro]], seen from space
File:Rio-cauto-cuba.JPG|Meanders: sinuous path of [[Cauto River|Rio Cauto]], Cuba
File:Jiangxia-snake-9704 (cropped).jpg|Meanders: sinuous snake crawling
File:Diplora strigosa (Symmetrical Brain Coral) closeup.jpg|Meanders: symmetrical [[brain coral]], ''Diploria strigosa''
</gallery>

===Waves, dunes ===

[[Wave]]s are disturbances that carry energy as they move. [[Mechanical wave]]s propagate through a medium – air or water, making it [[Oscillation|oscillate]] as they pass by.<ref>{{cite book |last=French |first=A. P. |title=Vibrations and Waves |publisher=Nelson Thornes |year=1971}}</ref> [[Wind wave]]s are sea [[surface wave]]s that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour can be predicted with wind wave models.<ref>{{cite conference |last=Tolman |first=H. L. |title=Practical wind wave modeling |book-title=CBMS Conference Proceedings on Water Waves: Theory and Experiment |date=2008 |conference=Howard University, USA, 13–18 May 2008 |publisher=World Scientific Publications |url=http://polar.ncep.noaa.gov/mmab/papers/tn270/Howard_08.pdf |editor-first=M.F. |editor-last=Mahmood}}</ref> As waves in water or wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create [[dune]]s, sometimes in extensive dune fields as in the [[Taklamakan]] desert. Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes.<ref>{{cite web |url=http://pubs.usgs.gov/gip/deserts/dunes/ |title=Types of Dunes |website=USGS |date=29 October 1997 |access-date=May 2, 2012}}</ref>

[[Barchan]]s or crescent dunes are produced by wind acting on desert sand; the two horns of the crescent and the [[slip face]] point downwind. Sand blows over the upwind face, which stands at about 15 degrees from the horizontal, and falls onto the slip face, where it accumulates up to the [[angle of repose]] of the sand, which is about 35 degrees. When the slip face exceeds the angle of repose, the sand [[avalanche]]s, which is a [[nonlinear]] behaviour: the addition of many small amounts of sand causes nothing much to happen, but then the addition of a further small amount suddenly causes a large amount to avalanche.<ref>{{cite book |last1=Strahler |first1=A. |last2=Archibold |first2=O. W. |title=Physical Geography: Science and Systems of the Human Environment |publisher=John Wiley |edition=4th |year=2008 |page=442}}</ref> Apart from this nonlinearity, barchans behave rather like [[soliton|solitary waves]].<ref>{{cite journal |last1=Schwämmle |first1=V. |last2=Herrman |first2=H. J. |title=Solitary wave behaviour of sand dunes |journal=Nature |volume=426 |date=11 December 2003 |issue=6967 |pages=619–620 |doi=10.1038/426619a |pmid=14668849 |bibcode=2003Natur.426..619S |s2cid=688445}}</ref>

<gallery mode=packed class="center">
File:Boelge stor.jpg|Waves: breaking wave in a ship's wake 
File:Taklimakanm.jpg|Dunes: sand dunes in [[Taklamakan]] desert, from space
File:Barchan.jpg|Dunes: [[barchan]] crescent sand dune
File:1969 Afghanistan (Sistan) wind ripples.tiff|Wind [[Capillary wave|ripples]] with [[dislocation]]s in [[Sistan]], Afghanistan
</gallery>

===Bubbles, foam===

A [[soap bubble]] forms a [[sphere]], a surface with minimal area ([[minimal surface]]) — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one.{{sfn|Stewart|2001|pages=108–109}}

A [[foam]] is a mass of bubbles; foams of different materials occur in nature. Foams composed of [[soap film]]s obey [[Plateau's laws]], which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the [[tetrahedron|tetrahedral]] angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant [[mean curvature|average curvature]] at every point. For example, a film may remain nearly flat on average by being curved up in one direction (say, left to right) while being curved downwards in another direction (say, front to back).{{sfn|Ball|2009a|page=68}}<ref>{{cite journal |last1=Almgren |first1=Frederick J. Jr. |last2=Taylor |first2=Jean E. |author2-link=Jean E. Taylor |title=The geometry of soap films and soap bubbles |journal=Scientific American |issue=235 |pages=82–93 |date=July 1976|volume=235 |doi=10.1038/scientificamerican0776-82 |bibcode=1976SciAm.235a..82A }}</ref> Structures with minimal surfaces can be used as tents.

At the scale of living [[cell (biology)|cells]], foam patterns are common; [[radiolarian]]s, [[sponge]] [[spicule (sponge)|spicule]]s, [[silicoflagellate]] [[exoskeleton]]s and the calcite skeleton of a [[sea urchin]], ''[[Cidaris rugosa]]'', all resemble mineral casts of Plateau foam boundaries.{{sfn|Ball|2009a|pages=96–101}}<ref>{{cite web |url=http://www.microscopy-uk.org.uk/mag/indexmag.html?http://www.microscopy-uk.org.uk/mag/artfeb05/cbdiatoms.html |title=Geometry and Pattern in Nature 3: The holes in radiolarian and diatom tests |website=Microscopy-UK |date=February 2005 |access-date=28 May 2012 |author=Brodie, Christina}}</ref> The skeleton of the [[Radiolarian]], ''Aulonia hexagona'', a beautiful marine form drawn by [[Ernst Haeckel]], looks as if it is a sphere composed wholly of hexagons, but this is mathematically impossible. The [[Euler characteristic]] states that for any [[convex polyhedron]], the number of faces plus the number of vertices (corners) equals the number of edges plus two. A result of this formula is that any closed polyhedron of hexagons has to include exactly 12 pentagons, like a [[Euler characteristic#Soccer ball|soccer ball]], [[Buckminster Fuller]] [[geodesic dome]], or [[fullerene]] molecule. This can be visualised by noting that a mesh of hexagons is flat like a sheet of chicken wire, but each pentagon that is added forces the mesh to bend (there are fewer corners, so the mesh is pulled in).{{sfn|Ball|2009a|pages=51–54}}

<gallery mode=packed class="center">
File:Foam - big.jpg|[[Foam]] of [[soap bubble]]s: four edges meet at each vertex, at angles close to 109.5°, as in two C-H bonds in [[methane]].
File:Haeckel Cyrtoidea.jpg|[[Radiolaria]] drawn by [[Ernst Haeckel|Haeckel]] in his ''Kunstformen der Natur'' (1904).
File:Haeckel Spumellaria.jpg|Haeckel's [[Spumellaria]]; the skeletons of these Radiolaria have foam-like forms.
File:C60 Molecule.svg|[[Buckminsterfullerene]] C<sub>60</sub>: [[Richard Smalley]] and colleagues synthesised the [[fullerene]] molecule in 1985.
File:3D_model_of_brochosome.jpg|[[Brochosomes]] (secretory [[microparticles]] produced by [[leafhoppers]]) often approximate [[fullerene]] geometry.
File:Equal spheres in a plane.tif|Equal spheres (gas bubbles) in a surface foam
File:CircusTent02.jpg|Circus tent approximates a minimal surface.
</gallery>

=== Tessellations ===
{{main|Tessellation}}

[[Tessellation]]s are patterns formed by repeating [[tile]]s all over a flat surface. There are 17 [[wallpaper group]]s of tilings.<ref name=armstrong>{{cite book |last=Armstrong |first=M.A. |title=Groups and Symmetry |url=https://archive.org/details/groupssymmetry0000arms |url-access=registration |date=1988 |publisher=Springer-Verlag |location=New York}}</ref> While common in art and design, exactly repeating tilings are less easy to find in living things. The cells in the paper nests of social [[wasp]]s, and the wax cells in [[honeycomb]] built by honey bees are well-known examples. Among animals, bony fish, reptiles or the [[pangolin]], or fruits like the [[salak]] are protected by overlapping scales or [[osteoderms]], these form more-or-less exactly repeating units, though often the scales in fact vary continuously in size. Among flowers, the snake's head fritillary, ''[[Fritillaria meleagris]]'', have a tessellated chequerboard pattern on their petals. The structures of [[minerals]] provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a [[crystal]], defined by [[crystal structure]], [[crystal system]], and [[point group]]; for example, there are exactly 14 [[Bravais lattices]] for the 7 lattice systems in three-dimensional space.<ref name="Solid State Physics 2010">Hook, J. R.; Hall, H. E. ''Solid State Physics'' (2nd Edition). Manchester Physics Series, John Wiley & Sons, 2010. {{isbn|978-0-471-92804-1}}</ref>

<gallery mode=packed class="center">
File:Halite-249324 (3x4).jpg|Crystals: cube-shaped crystals of [[halite]] (rock salt); [[cubic crystal system]], isometric hexoctahedral crystal symmetry
File:Kin selection, Honey bees.jpg|Arrays: [[honeycomb]] is a natural [[tessellation]]
File:Wismut Kristall und 1cm3 Wuerfel.jpg|[[Bismuth]] [[hopper crystal]] illustrating the stairstep [[crystal habit]].
File:Fritillaria-meleagris-blomst.JPG|Tilings: tessellated flower of snake's head fritillary, ''[[Fritillaria meleagris]]'' 
File:Scale Common Roach.JPG|Tilings: overlapping scales of common roach, ''[[Rutilus rutilus]]''
File:Salak fruits Salacca zalacca.jpg|Tilings: overlapping scales of snakefruit or [[salak]], ''Salacca zalacca''
File:Tessellated Pavement Sunrise Landscape.jpg|[[Tessellated pavement]]: a rare rock formation on the [[Tasman Peninsula]]
</gallery>

=== Cracks ===

[[Fracture|Cracks]] are linear openings that form in materials to relieve [[Stress (mechanics)|stress]]. When an [[Elasticity (physics)|elastic]] material stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly in all directions, creating cracks with 120 degree joints, so three cracks meet at a node. Conversely, when an inelastic material fails, straight cracks form to relieve the stress. Further stress in the same direction would then simply open the existing cracks; stress at right angles can create new cracks, at 90 degrees to the old ones. Thus the pattern of cracks indicates whether the material is elastic or not.{{sfn|Stevens|1974|page=207}} In a tough fibrous material like oak tree bark, cracks form to relieve stress as usual, but they do not grow long as their growth is interrupted by bundles of strong elastic fibres. Since each species of tree has its own structure at the levels of cell and of molecules, each has its own pattern of splitting in its bark.{{sfn|Stevens|1974|page=208}}

<gallery mode=packed class="center">
File:Old Pottery surface with 90 degree cracks.jpg|Old pottery surface, white glaze with mainly 90° cracks
File:Cracked earth in the Rann of Kutch.jpg|Drying inelastic mud in the [[Rann of Kutch]] with mainly 90° cracks
Veined Gabbro with 90 degree cracks, Sgurr na Stri, Skye.jpg|Veined [[gabbro]] with 90° cracks, near [[List of Marilyns on Scottish islands|Sgurr na Stri]], [[Skye]]
File:Drying mud with 120 degree cracks, Sicily.jpg|Drying elastic mud in [[Sicily]] with mainly 120° cracks
File:Causeway-code poet-4.jpg|Cooled [[basalt]] at [[Giant's Causeway]]. Vertical mainly 120° cracks giving hexagonal columns
File:Palm tree bark pattern.jpg|Palm trunk with branching vertical cracks (and horizontal leaf scars)
</gallery>

===Spots, stripes===

Leopards and ladybirds are spotted; angelfish and zebras are striped.{{sfn|Ball|2009a|pages=156–158}} These patterns have an [[evolution]]ary explanation: they have [[Function (biology)|functions]] which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is [[camouflage]];<ref name=DarwinCh4/> for instance, a [[leopard]] that is harder to see catches more prey. Another function is [[Signalling theory|signalling]]<ref name=Wickler/> — for instance, a [[ladybird]] is less likely to be attacked by [[predator]]y birds that hunt by sight, if it has bold warning colours, and is also [[Aposematism|distastefully bitter or poisonous]], or [[Mimicry|mimics]] other distasteful insects. A young bird may see a warning patterned insect like a ladybird and try to eat it, but it will only do this once; very soon it will spit out the bitter insect; the other ladybirds in the area will remain undisturbed. The young leopards and ladybirds, inheriting [[gene]]s that somehow create spottedness, survive. But while these evolutionary and functional arguments explain why these animals need their patterns, they do not explain how the patterns are formed.{{sfn|Ball|2009a|pages=156–158}}

<gallery mode=packed class="center">
File:Dirce Beauty Colobura dirce.jpg|Dirce beauty butterfly, ''[[Colobura dirce]]''
File:Equus grevyi (aka).jpg|[[Grevy's zebra]], ''Equus grevyi''
File:Angelfish Nick Hobgood.jpg|[[Royal angelfish]], ''Pygoplites diacanthus''
File:Leopard africa.jpg|[[Leopard]], ''Panthera pardus pardus''
File:Georgiy Jacobson - Beetles Russia and Western Europe - plate 24.jpg|Array of [[ladybird]]s by [[Georgij Georgiewitsch Jacobson|G.G. Jacobson]]
File:Sepia officinalis Cuttlefish striped breeding pattern.jpg|Breeding pattern of [[cuttlefish]], ''Sepia officinalis''
</gallery>

==Pattern formation==
{{main|Pattern formation}}

Alan Turing,<ref name=Turing/> and later the mathematical biologist [[James D. Murray|James Murray]],<ref name="Murray2013">{{cite book |last=Murray |first=James D. |title=Mathematical Biology |url=https://books.google.com/books?id=K3LmCAAAQBAJ&pg=PA436 |date=9 March 2013 |publisher=Springer Science & Business Media |isbn=978-3-662-08539-4 |pages=436–450}}</ref> described a mechanism that spontaneously creates spotted or striped patterns: a [[reaction–diffusion system]].{{sfn|Ball|2009a|pages=159–167}} The cells of a young organism have genes that can be switched on by a chemical signal, a [[morphogen]], resulting in the growth of a certain type of structure, say a darkly pigmented patch of skin. If the morphogen is present everywhere, the result is an even pigmentation, as in a black leopard. But if it is unevenly distributed, spots or stripes can result. Turing suggested that there could be [[feedback]] control of the production of the morphogen itself. This could cause continuous fluctuations in the amount of morphogen as it diffused around the body. A second mechanism is needed to create [[standing wave]] patterns (to result in spots or stripes): an inhibitor chemical that switches off production of the morphogen, and that itself diffuses through the body more quickly than the morphogen, resulting in an activator-inhibitor scheme. The [[Belousov–Zhabotinsky reaction]] is a non-biological example of this kind of scheme, a [[chemical oscillator]].{{sfn|Ball|2009a|pages=159–167}}

Later research has managed to create convincing models of patterns as diverse as zebra stripes, giraffe blotches, jaguar spots (medium-dark patches surrounded by dark broken rings) and ladybird shell patterns (different geometrical layouts of spots and stripes, see illustrations).{{sfn|Ball|2009a|pages=168–180}} [[Richard Prum]]'s activation-inhibition models, developed from Turing's work, use six variables to account for the observed range of nine basic within-feather pigmentation patterns, from the simplest, a central pigment patch, via concentric patches, bars, chevrons, eye spot, pair of central spots, rows of paired spots and an array of dots.{{sfn|Rothenberg|2011|pages=93–95}}<ref name=Prum/> More elaborate models simulate complex feather patterns in the guineafowl ''[[Numida meleagris]]'' in which the individual feathers feature transitions from bars at the base to an array of dots at the far (distal) end. These require an oscillation created by two inhibiting signals, with interactions in both space and time.<ref name=Prum>{{cite journal |url=http://www.yale.edu/eeb/prum/pdf/Prum%20&%20Williamson%202002.pdf |title=Reaction–diffusion models of within-feather pigmentation patterning |last1=Prum |first1=Richard O. |author1-link=Richard Prum |last2=Williamson |first2=Scott |journal=Proceedings of the Royal Society of London B |date=2002 |volume=269 |issue=1493 |pages=781–792 |doi=10.1098/rspb.2001.1896|pmid=11958709 |pmc=1690965}}</ref>

Patterns can form for other reasons in the [[patterned vegetation|vegetated landscape]] of [[tiger bush]]<ref name=TigerBush>{{cite book |title=Banded vegetation patterning in arid and semiarid environments |publisher=Springer-Verlag |last1=Tongway |first1=D. J. |author2=Valentin, C. |author3=Seghieri, J. |date=2001 |location=New York}}</ref> and [[fir wave]]s.<ref name=FirWave>{{cite web |url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html |title=Fir Waves: Regeneration in New England Conifer Forests |website=TIEE |date=22 February 2004 |access-date=26 May 2012 |author=D'Avanzo, C.}}</ref> Tiger bush stripes occur on arid slopes where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation effectively collects the rainwater from the bare zone immediately above it.<ref name=TigerBush/> Fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow, protected by the wind shadow of the remaining tall trees.<ref name=FirWave/> Natural patterns are sometimes formed by animals, as in the [[Mima mounds]] of the Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of [[pocket gopher]]s,<ref name=BBC>{{cite web |last=Morelle |first=Rebecca |author-link=Rebecca Morelle |title='Digital gophers' solve Mima mound mystery |url=https://www.bbc.co.uk/news/science-environment-25259133 |website=BBC News |access-date=9 December 2013|date=2013-12-09 }}</ref> while the so-called [[fairy circle]]s of Namibia appear to be created by the interaction of competing groups of sand termites, along with competition for water among the desert plants.<ref>{{cite news |last1=Sample |first1=Ian |title=The secret of Namibia's 'fairy circles' may be explained at last |url=https://www.theguardian.com/science/2017/jan/18/the-secret-of-namibias-fairy-circles-may-be-explained-at-last |newspaper=[[The Guardian]] |access-date=18 January 2017 |date=18 January 2017}}</ref>

In permafrost soils with an active upper layer subject to annual freeze and thaw, [[patterned ground]] can form, creating circles, nets, [[ice wedge]] polygons, steps, and stripes. [[Thermal contraction]] causes shrinkage cracks to form; in a thaw, water fills the cracks, expanding to form ice when next frozen, and widening the cracks into wedges. These cracks may join up to form polygons and other shapes.<ref>{{cite web |title=Permafrost: Patterned Ground |url=http://permafrosttunnel.crrel.usace.army.mil/permafrost/patterned_ground.html |website=US Army Corps of Engineers |access-date=17 February 2015 |archive-url=https://web.archive.org/web/20150307013043/http://permafrosttunnel.crrel.usace.army.mil/permafrost/patterned_ground.html |archive-date=7 March 2015 |url-status=dead }}</ref>

The [[gyrification|fissured pattern]] that develops on vertebrate brains is caused by a physical process of constrained expansion dependent on two geometric parameters: relative tangential cortical expansion and relative thickness of the [[cerebellar cortex|cortex]]. Similar patterns of [[Gyrus|gyri]] (peaks) and [[Sulcus (neuroanatomy)|sulci]] (troughs) have been demonstrated in models of the brain starting from smooth, layered gels, with the patterns caused by compressive mechanical forces resulting from the expansion of the outer layer (representing the cortex) after the addition of a solvent. Numerical models in computer simulations support natural and experimental observations that the surface folding patterns increase in larger brains.<ref name="Scientific American">{{cite web |url=https://www.scientificamerican.com/article/human-brain-s-bizarre-folding-pattern-re-created-in-a-vat/ |last1=Ghose |first1=Tia |title=Human Brain's Bizarre Folding Pattern Re-Created in a Vat |website=[[Scientific American]] |access-date=5 April 2018}}</ref><ref name="PNAS">{{cite journal |last1=Tallinen |first1=Tuoma|last2=Chung |first2=Jun Young |last3=Biggins |first3= John S. |last4=Mahadevan |first4=L. |title=Gyrification from constrained cortical expansion |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=111 |issue=35 |pages=12667–12672 |date=2014 |doi=10.1073/pnas.1406015111 |pmid=25136099 |arxiv=1503.03853 |pmc=4156754 |bibcode=2014PNAS..11112667T|doi-access=free}}</ref>

<gallery mode=packed class="center">
File:Giant Puffer fish skin pattern.JPG|[[Giant pufferfish]], ''Tetraodon mbu''
File:Giant Pufferfish skin pattern detail.jpg|Detail of giant pufferfish skin pattern
File:Belousov-Zhabotinsky Reaction Simulation Snapshot.jpg|Snapshot of simulation of [[Belousov–Zhabotinsky reaction]]
File:Pintade de Numidie.jpg|Helmeted guineafowl, ''[[Numida meleagris]]'', feathers transition from barred to spotted, both in-feather and across the bird
File:Tiger Bush Niger Corona 1965-12-31.jpg|Aerial view of a [[tiger bush]] [[plateau]] in [[Niger]]
File:Fir waves.jpg|[[Fir waves]] in [[White Mountains (New Hampshire)|White Mountains]], [[New Hampshire]]
File:Melting pingo wedge ice.jpg|[[Patterned ground]]: a melting [[pingo]] with surrounding [[ice wedge]] polygons near [[Tuktoyaktuk]], Canada
File:Fairy circles namibia.jpg|[[Fairy circle]]s in the Marienflusstal area in [[Namibia]]
File:02 1 facies dorsalis cerebri.jpg|Human brain (superior view) exhibiting patterns of [[Gyrus|gyri]] and [[Sulcus (neuroanatomy)|sulci]]
</gallery>

==See also==

* [[Developmental biology]]
* [[Emergence]]
* [[Evolutionary history of plants]]
* [[Mathematics and art]]
* [[Morphogenesis]]
* [[Pattern formation]]
* [[Widmanstätten pattern]]

==References==
'''Footnotes'''
{{notelist}}

'''Citations'''
{{reflist|30em}}

===Bibliography===

'''Pioneering authors'''
{{Wikisourcelang|la|Liber abbaci}}
* [[Fibonacci|Fibonacci, Leonardo]]. ''[[Liber Abaci]]'', 1202.
** ———— translated by Sigler, Laurence E. ''Fibonacci's Liber Abaci''. Springer, 2002.
* [[Ernst Haeckel|Haeckel, Ernst]]. ''[[Kunstformen der Natur]]'' (Art Forms in Nature), 1899–1904.
* [[D'Arcy Wentworth Thompson|Thompson, D'Arcy Wentworth]]. ''[[On Growth and Form]]''. Cambridge, 1917.

'''General books'''
* Adam, John A. [http://press.princeton.edu/titles/7686.html ''Mathematics in Nature: Modeling Patterns in the Natural World'']. [[Princeton University Press]], 2006.
* {{cite book |author-link=Philip Ball |last=Ball |first=Philip |title=Nature's Patterns: a tapestry in three parts. 1: Shapes |publisher=Oxford University Press |year=2009a}}
* {{cite book |last=Ball |first=Philip |title=Nature's Patterns: a tapestry in three parts. 2: Flow |publisher=Oxford University Press |year=2009b |ref=none}}
* {{cite book |last=Ball |first=Philip |title=Nature's Patterns: a tapestry in three parts. 3. Branches |publisher=Oxford University Press |year=2009c |ref=none}}
* Ball, Philip. ''Patterns in Nature''. Chicago, 2016.
* [[Pat Murphy (writer)|Murphy, Pat]] and Neill, William. ''By Nature's Design''. [[Chronicle Books]], 1993.
* {{cite book |author-link=David Rothenberg |last=Rothenberg |first=David |title=Survival of the Beautiful: Art, Science and Evolution |publisher=[[Bloomsbury Press]] |year=2011 }}
* {{cite book |last=Stevens |first=Peter S. |title=Patterns in Nature |publisher=[[Little, Brown & Co]] |year=1974 }}
* {{cite book|author-link=Ian Stewart (mathematician)|last=Stewart |first=Ian |title=What Shape is a Snowflake? Magical Numbers in Nature |publisher=[[Weidenfeld & Nicolson]] |year=2001}}

'''Patterns from nature (as art)'''
* Edmaier, Bernard. ''Patterns of the Earth''. [[Phaidon Press]], 2007.
* Macnab, Maggie. ''Design by Nature: Using Universal Forms and Principles in Design''. New Riders, 2012.
* Nakamura, Shigeki. ''Pattern Sourcebook: 250 Patterns Inspired by Nature.''. Books 1 and 2. Rockport, 2009.
* O'Neill, Polly. ''Surfaces and Textures: A Visual Sourcebook''. Black, 2008.
* [[Porter, Eliot]], and [[James Gleick|Gleick, James]]. ''Nature's Chaos''. [[Viking Penguin]], 1990.

==External links==
* [https://web.archive.org/web/20160210085824/http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/ Fibonacci Numbers and the Golden Section]
* [https://web.archive.org/web/20120507072405/http://www.math.smith.edu/phyllo/ Phyllotaxis: an Interactive Site for the Mathematical Study of Plant Pattern Formation]

{{Patterns in nature}}
{{Aesthetics}}
{{Authority control}}

[[Category:Applied mathematics]]
[[Category:History of science]]
[[Category:Nature]]
[[Category:Pattern formation]]
[[Category:Patterns]]
[[Category:Recreational mathematics]]