Patterns in nature

Natural patterns form as wind blows sand in the dunes of theNamib Desert. The crescent shaped dunes and the ripples ontheir surfaces repeat wherever there are suitable conditions.

Patterns in nature are visible regularities of form foundin the natural world. These patterns recur in differ-ent contexts and can sometimes be modelled mathemati-cally. Natural patterns include symmetries, trees, spirals,meanders, waves, foams, arrays, cracks and stripes.[1]Early Greek philosophers studied pattern, with Plato,Pythagoras and Empedocles attempting to explain orderin nature. The modern understanding of visible patternsdeveloped gradually over time.In the 19th century, Belgian physicist Joseph Plateau ex-

Patterns of the veiled chameleon, Chamaeleo calyptratus,evolved for camouflage and to signal mood and breeding con-dition.

amined soap films, leading him to formulate the con-cept of a minimal surface. German biologist and artistErnst Haeckel painted hundreds of marine organisms toemphasise their symmetry. Scottish biologist D'ArcyThompson pioneered the study of growth patterns inboth plants and animals, showing that simple equa-tions could explain spiral growth. In the 20th cen-tury, British mathematician Alan Turing predicted mech-anisms of morphogenesis which give rise to patterns ofspots and stripes. Hungarian biologist Aristid Linden-mayer and French American mathematician Benoît Man-delbrot showed how the mathematics of fractals couldcreate plant growth patterns.Mathematics, physics and chemistry can explain patternsin nature at different levels. Patterns in living things areexplained by the biological processes of natural selectionand sexual selection. Studies of pattern formation makeuse of computer models to simulate a wide range of pat-terns.

1

2 1 HISTORY

Fibonacci patterns occur widely in plant structures including thiscone of Queen sago, Cycas circinalis

1 History

Early Greek philosophers attempted to explain order innature, anticipating modern concepts. Plato (c 427 – c347 BC) — looking only at his work on natural patterns— argued for the existence of universals. He consid-ered these to consist of ideal forms (εἶδος eidos: “form”)of which physical objects are never more than imperfectcopies. Thus, a flower may be roughly circular, but itis never a perfect mathematical circle.[2] Pythagoras ex-plained patterns in nature like the harmonies of musicas arising from number, which he took to be the basicconstituent of existence.[3] Empedocles to an extent an-ticipated Darwin's evolutionary explanation for the struc-tures of organisms.[4]

In 1202, Leonardo Fibonacci (c 1170 – c 1250) intro-duced the Fibonacci number sequence to the westernworld with his book Liber Abaci.[5] Fibonacci gave an(unrealistic) biological example, on the growth in num-bers of a theoretical rabbit population.[6] In 1917, D'ArcyWentworth Thompson (1860–1948) published his book

On Growth and Form. His description of phyllotaxis andthe Fibonacci sequence, the mathematical relationshipsin the spiral growth patterns of plants, is classic. Heshowed that simple equations could describe all the ap-parently complex spiral growth patterns of animal hornsand mollusc shells.[7]

Belgian physicist Joseph Plateau (1801–1883) formu-lated the mathematical problem of the existence of aminimal surface with a given boundary, which is nownamed after him. He studied soap films intensively,formulating Plateau’s laws which describe the structuresformed by films in foams.[8]

Ernst Haeckel (1834–1919) painted beautiful illustra-tions of marine organisms, in particular Radiolaria, em-phasising their symmetry to support his faux-Darwiniantheories of evolution.[9]

The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in1885.[10]

Crocodilus porosus Notosuchus terrestrisC. americanus

D'Arcy Thompson pioneered the study of growth and form

In 1952, Alan Turing (1912–1954), better known for hiswork on computing and codebreaking, wrote The Chem-ical Basis of Morphogenesis, an analysis of the mech-anisms that would be needed to create patterns in liv-ing organisms, in the process called morphogenesis.[11]He predicted oscillating chemical reactions, in particu-lar the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms can, Turing suggested, generatepatterns of stripes and spots in animals, and contributeto the spiral patterns seen in plant phyllotaxis.[12]

In 1968, Hungarian theoretical biologist Aristid Linden-mayer (1925–1989) developed the L-system, a formalgrammar which can be used to model plant growthpatterns in the style of fractals.[13] L-systems have

3

an alphabet of symbols that can be combined usingproduction rules to build larger strings of symbols, and amechanism for translating the generated strings into ge-ometric structures. In 1975, after centuries of slow de-velopment of the mathematics of patterns by GottfriedLeibniz, Georg Cantor, Helge von Koch, Wacław Sier-piński and others, Benoît Mandelbrot wrote a famous pa-per, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, crystallising math-ematical thought into the concept of the fractal.[14]

2 Causes

Composite patterns: aphids and newly born young in arraylikeclusters on Sycamore leaf, divided into polygons by veins, whichare avoided by the young aphids

Living things like orchids, hummingbirds, and thepeacock’s tail have abstract designs with a beauty of form,pattern and colour that artists struggle to match.[15] Thebeauty that people perceive in nature has causes at differ-ent levels, notably in the mathematics that governs what

patterns can physically form, and among living things inthe effects of natural selection, that govern how patternsevolve.[16]

Mathematics seeks to discover and explain abstract pat-terns or regularities of all kinds.[17][18] Visual patterns innature find explanations in chaos theory, fractals, log-arithmic spirals, topology and other mathematical pat-terns. For example, L-systems form convincing modelsof different patterns of tree growth.[13]

The growth patterns of certain trees resemble these Lindenmayersystem fractals.

The laws of physics apply the abstractions of mathemat-ics to the real world, often as if it were perfect. For ex-ample a crystal is perfect when it has no structural defectssuch as dislocations and is fully symmetric. Exact mathe-matical perfection can only approximate real objects.[19]Visible patterns in nature are governed by physical laws;for example, meanders can be explained using fluid dy-namics.In biology, natural selection, can cause the developmentof patterns in living things for several reasons, includingcamouflage,[20] sexual selection,[20] and different kindsof signalling, including mimicry[21] and cleaning sym-biosis.[22] In plants, the shapes, colours, and patterns offlowers like the lily have evolved to optimise insect polli-nation (other plants may be pollinated by wind, birds, orbats). European honey bees and other pollinating insectsare attracted to flowers by a radial pattern of colours andstripes (some visible only in ultraviolet light) that serveas nectar guides that can be seen at a distance; by scent;and by rewards of sugar-rich nectar and edible pollen.[23]

4 3 TYPES OF PATTERN

3 Types of pattern

3.1 Symmetry

Further information: Symmetry in biology, Floralsymmetry and Crystal symmetry

Symmetry is pervasive in living things. Animals mainlyhave bilateral or mirror symmetry, as do the leaves ofplants and some flowers such as orchids.[24] Plants oftenhave radial or rotational symmetry, as do many flowersand some groups of animals such as sea anemones. Five-fold symmetry is found in the echinoderms, the group thatincludes starfish, sea urchins, and sea lilies.[25]

Among non-living things, snowflakes have strikingsixfold symmetry: each flake is unique, its structure form-ing a record of the varying conditions during its crystalli-sation, with nearly the same pattern of growth on eachof its six arms.[26] Crystals in general have a variety ofsymmetries and crystal habits; they can be cubic or oc-tahedral, but true crystals cannot have fivefold symmetry(unlike quasicrystals).[27] Rotational symmetry is foundat different scales among non-living things including thecrown-shaped splash pattern formed when a drop fallsinto a pond,[28] and both the spheroidal shape and ringsof a planet like Saturn.[29]

Symmetry has a variety of causes. Radial symmetry suitsorganisms like sea anemones whose adults do not move:food and threats may arrive from any direction. But an-imals that move in one direction necessarily have upperand lower sides, head and tail ends, and therefore a leftand a right. The head becomes specialised with a mouthand sense organs (cephalisation), and the body becomesbilaterally symmetric (though internal organs need notbe).[30] More puzzling is the reason for the fivefold (pen-taradiate) symmetry of the echinoderms. Early echino-derms were bilaterally symmetrical, as their larvae stillare. Sumrall andWray argue that the loss of the old sym-metry had both developmental and ecological causes.[31]

• Animals often show mirror or bilateral symmetry,like this tiger.

• Echinoderms like this starfish have fivefold symme-try.

• Fivefold symmetry can be seen in many flowers andsome fruits like this medlar.

• Snowflakes have sixfold symmetry.

• Each snowflake is unique but symmetrical.

• Fluorite showing cubic crystal habit

• Water splash approximates radial symmetry.

• Garnet showing rhombic dodecahedral crystal habit

• Volvox has spherical symmetry.

• Sea anemones have rotational symmetry.

3.2 Trees, fractals

Fractals are infinitely self-similar, iterated mathemati-cal constructs having fractal dimensions.[14][32][33] Infi-nite iteration is not possible in nature so all 'fractal' pat-terns are only approximate. For example, the leaves offerns and umbellifers (Apiaceae) are only self-similar(pinnate) to 2, 3 or 4 levels. Fern-like growth pat-terns occur in plants and in animals including bryozoa,corals, hydrozoa like the air fern, Sertularia argentea,and in non-living things, notably electrical discharges.Lindenmayer system fractals can model different patternsof tree growth by varying a small number of parame-ters including branching angle, distance between nodes orbranch points (internode length), and number of branchesper branch point.[13]

Fractal-like patterns occur widely in nature, in phe-nomena as diverse as clouds, river networks, geologicfault lines, mountains, coastlines,[14] animal coloration,snow flakes,[34] crystals,[35] blood vessel branching,[36]and ocean waves.[37]

• Leaf of Cow Parsley, Anthriscus sylvestris, is 2- or3-pinnate, not infinite

• Fractal spirals: Romanesco broccoli showing self-similar form

• Angelica flowerhead, a sphere made of spheres(self-similar)

• Trees: Lichtenberg figure: high voltage dielectricbreakdown in an acrylic polymer block

• Trees: dendritic Copper crystals (in microscope)

3.4 Chaos, flow, meanders 5

3.3 Spirals

Spirals are common in plants and in some animals, no-tably molluscs. For example, in the nautilus, a cephalo-pod mollusc, each chamber of its shell is an approximatecopy of the next one, scaled by a constant factor and ar-ranged in a logarithmic spiral.[38] Given a modern under-standing of fractals, a growth spiral can be seen as a spe-cial case of self-similarity.[39]

Plant spirals can be seen in phyllotaxis, the arrange-ment of leaves on a stem, and in the arrangement(parastichy[40]) of other parts as in composite flowerheads and seed heads like the sunflower or fruit struc-tures like the pineapple[41]:337 and snake fruit, as well as inthe pattern of scales in pine cones, where multiple spiralsrun both clockwise and anticlockwise. These arrange-ments have explanations at different levels – mathemat-ics, physics, chemistry, biology – each individually cor-rect, but all necessary together.[42] Phyllotaxis spirals canbe generated mathematically from Fibonacci ratios: theFibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (each subse-quent number being the sum of the two preceding ones).For example, when leaves alternate up a stem, one ro-tation of the spiral touches two leaves, so the pattern orratio 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.[43] In disc phyl-lotaxis as in the sunflower and daisy, the florets are ar-ranged in Fermat’s spiral with Fibonacci numbering, atleast when the flowerhead ismature so all the elements arethe same size. Fibonacci ratios approximate the goldenangle, 137.508°, which governs the curvature of Fermat’sspiral.[44]

From the point of view of physics, spirals are lowest-energy configurations[45] which emerge spontaneouslythrough self-organizing processes in dynamic systems.[46]From the point of view of chemistry, a spiral can begenerated by a reaction-diffusion process, involving bothactivation and inhibition. Phyllotaxis is controlled byproteins that manipulate the concentration of the planthormone auxin, which activates meristem growth, along-side other mechanisms to control the relative angle ofbuds around the stem.[47] From a biological perspective,arranging leaves as far apart as possible in any given spaceis favoured by natural selection as it maximises access toresources, especially sunlight for photosynthesis.[41]

• Fibonacci spiral

• Bighorn sheep, Ovis canadensis

• Spirals: phyllotaxis of spiral aloe, Aloe polyphylla

• Nautilus shell’s logarithmic growth spiral

• Fermat’s spiral: seed head of sunflower, Helianthusannuus

• Multiple Fibonacci spirals: red cabbage in cross sec-tion

• Gastropod mollusc shell, Trochoidea liebetruti,showing how opening moves around, outward, anddownwards as it grows

3.4 Chaos, flow, meanders

In mathematics, a dynamical system is chaotic if itis (highly) sensitive to initial conditions (the so-calledextquotedblbutterfly effect extquotedbl[48]), which re-quires the mathematical properties of topological mixingand dense periodic orbits.[49]

Alongside fractals, chaos theory ranks as an essentiallyuniversal influence on patterns in nature. There is a re-lationship between chaos and fractals—the strange at-tractors in chaotic systems have a fractal dimension.[50]Some cellular automata, simple sets of mathematicalrules that generate patterns, have chaotic behaviour, no-tably Stephen Wolfram's Rule 30.[51]

Vortex streets are zigzagging patterns of whirling vorticescreated by the unsteady separation of flow of a fluid, mostoften air or water, over obstructing objects.[52] Smooth(laminar) flow starts to break up when the size of the ob-struction or the velocity of the flow become large enoughcompared to the viscosity of the fluid.Meanders are sinuous bends in rivers or other channels,which form as a fluid, most often water, flows aroundbends. As soon as the path is slightly curved, the sizeand curvature of each loop increases as helical flow dragsmaterial like sand and gravel across the river to the insideof the bend. The outside of the loop is left clean and un-protected, so erosion accelerates, further increasing themeandering in a powerful positive feedback loop.[53]

• Chaos: shell of gastropod mollusc the cloth of goldcone, Conus textile, resembles Rule 30 cellular au-tomaton

• Chaos: vortex street of clouds

6 3 TYPES OF PATTERN

• Meanders: dramatic meander scars and oxbow lakesin the broad flood plain of the Rio Negro, seen fromspace

• Meanders: sinuous path of Rio Cauto, Cuba

• Meanders: sinuous snake crawling

• Meanders: symmetrical Brain Coral, Diploriastrigosa

3.5 Waves, dunes

Waves are disturbances that carry energy as they move.Mechanical waves propagate through a medium – air orwater, making it oscillate as they pass by.[54] Wind wavesare sea surface waves that create the characteristic chaoticpattern of any large body of water, though their statisticalbehaviour can be predicted with wind wave models.[55]As waves in water or wind pass over sand, they createpatterns of ripples. When winds blow over large bodiesof sand, they create dunes, sometimes in extensive dunefields as in the Taklamakan desert. Dunes may form arange of patterns including crescents, very long straightlines, stars, domes, parabolas, and longitudinal or Seif('sword') shapes.[56]

Barchans or crescent dunes are produced by wind act-ing on desert sand; the two horns of the crescent and theslip face point downwind. Sand blows over the upwindface, which stands at about 15 degrees from the horizon-tal, and falls on to the slip face, where it accumulates upto the angle of repose of the sand, which is about 35 de-grees. When the slip face exceeds the angle of repose,the sand avalanches, which is a nonlinear behaviour: theaddition of many small amounts of sand causes nothingmuch to happen, but then the addition of a further smallamount suddenly causes a large amount to avalanche.[57]Apart from this nonlinearity, barchans behave rather likesolitary waves.[58]

• Waves: breaking wave in a ship’s wake

• Dunes: sand dunes in Taklamakan desert, fromspace

• Dunes: barchan crescent sand dune

• Wind ripples with dislocations in Sistan,Afghanistan

3.6 Bubbles, foam

A soap bubble forms a sphere, a surface with minimalarea — the smallest possible surface area for the volumeenclosed. Two bubbles together form a more complexshape: the outer surfaces of both bubbles are spherical;these surfaces are joined by a third spherical surface asthe smaller bubble bulges slightly into the larger one.[8]

A foam is a mass of bubbles; foams of different materi-als occur in nature. Foams composed of soap films obeyPlateau’s laws, which require three soap films to meet ateach edge at 120° and four soap edges to meet at eachvertex at the tetrahedral angle of about 109.5°. Plateau’slaws further require films to be smooth and continuous,and to have a constant average curvature at every point.For example, a film may remain nearly flat on averageby being curved up in one direction (say, left to right)while being curved downwards in another direction (say,front to back).[59][60] Structures with minimal surfacescan be used as tents. Lord Kelvin identified the problemof the most efficient way to pack cells of equal volumeas a foam in 1887; his solution uses just one solid, thebitruncated cubic honeycomb with very slightly curvedfaces tomeet Plateau’s laws. No better solution was founduntil 1993 when Denis Weaire and Robert Phelan pro-posed the Weaire–Phelan structure; the Beijing NationalAquatics Center adapted the structure for their outer wallin the 2008 Summer Olympics.[61]

At the scale of living cells, foam patterns are common;radiolarians, sponge spicules, silicoflagellate exoskeletonsand the calcite skeleton of a sea urchin, Cidaris ru-gosa, all resemble mineral casts of Plateau foamboundaries.[62][63] The skeleton of the Radiolarian, Aulo-nia hexagona, a beautiful marine form drawn by Haeckel,looks as if it is a sphere composed wholly of hexagons,but this is mathematically impossible. The Euler charac-teristic states that for any convex polyhedron, the num-ber of faces plus the number of vertices (corners) equalsthe number of edges plus two. A result of this formula isthat any closed polyhedron of hexagons has to include ex-actly 12 pentagons, like a soccer ball, Buckminster Fullergeodesic dome, or fullerene molecule. This can be visu-alised by noting that a mesh of hexagons is flat like a sheetof chicken wire, but each pentagon that is added forcesthe mesh to bend (there are fewer corners, so the mesh ispulled in).[64]

• Foam of soap bubbles: 4 edges meet at each vertex,at angles close to 109.5°, as in two C-H bonds in

3.8 Cracks 7

methane.

• Radiolaria drawn by Haeckel in his Kunstformen derNatur (1904).

• Haeckel’s Spumellaria; the skeletons of these Radi-olaria have foam-like forms.

• Buckminsterfullerene C60: Richard Smalley andcolleagues synthesised the fullerene molecule in1985.

• Circus tent approximates a minimal surface.

• Beijing’s National Aquatics Center for the 2008Olympic games has a Weaire-Phelan surface.

3.7 Arrays, crystals, tilings

Tessellations are patterns formed by repeating tiles allover a flat surface. There are 17 wallpaper groups oftilings.[65] While common in art and design, exactly re-peating tilings are less easy to find in living things. Thecells in the paper nests of social wasps, and the wax cellsin honeycomb built by honey bees are well-known exam-ples. Among animals, bony fish, reptiles or the pangolin,or fruits like the Salak are protected by overlapping scalesor osteoderms, these form more-or-less exactly repeat-ing units, though often the scales in fact vary continu-ously in size. Among flowers, the Snake’s Head Fritillary,Fritillaria meleagris, have a tessellated chequerboard pat-tern on their petals. The structures of minerals providegood examples of regularly repeating three-dimensionalarrays. Despite the hundreds of thousands of knownminerals, there are rather few possible types of arrange-ment of atoms in a crystal, defined by crystal structure,crystal system, and point group; for example, there areexactly 14 Bravais lattices for the 7 lattice systems inthree-dimensional space.[66]

• Crystals: cube-shaped crystals of halite (rock salt);cubic crystal system, Isometric hexoctahedral crys-tal symmetry

• Arrays: honeycomb is a natural tessellation

• Tilings: tessellated flower of Snake’s Head Fritil-lary, Fritillaria meleagris

• Tilings: overlapping scales of Common Roach,Rutilus rutilus

• Tilings: overlapping scales of snakefruit or salak,Salacca zalacca

• Tessellated pavement: a rare rock formation on theTasman Peninsula

3.8 Cracks

Cracks are linear openings that form in materials to re-lieve stress. When an elastic material stretches or shrinksuniformly, it eventually reaches its breaking strength andthen fails suddenly in all directions, creating cracks with120 degree joints, so three cracks meet at a node. Con-versely, when an inelastic material fails, straight cracksform to relieve the stress. Further stress in the same di-rection would then simply open the existing cracks; stressat right angles can create new cracks, at 90 degrees to theold ones. Thus the pattern of cracks indicates whetherthe material is elastic or not.[67] In a tough fibrous ma-terial like oak tree bark, cracks form to relieve stress asusual, but they do not grow long as their growth is in-terrupted by bundles of strong elastic fibres. Since eachspecies of tree has its own structure at the levels of celland of molecules, each has its own pattern of splitting inits bark.[68]

• Old pottery surface, white glaze with mainly 90°cracks

• Drying inelastic mud in the Rann of Kutch withmainly 90° cracks

• Veined gabbro with 90° cracks, near Sgurr na Stri,Skye

• Drying elasticmud in Sicily withmainly 120° cracks

• Cooled basalt at Giant’s Causeway. Vertical mainly120° cracks giving hexagonal columns

• Palm trunk with branching vertical cracks (and hor-izontal leaf scars)

3.9 Spots, stripes

Leopards and ladybirds are spotted; angelfish and ze-bras are striped.[69] These patterns have an evolutionaryexplanation: they have functions which increase thechances that the offspring of the patterned animal willsurvive to reproduce. One function of animal patterns is

8 3 TYPES OF PATTERN

camouflage;[20] for instance, a leopard that is harder tosee catches more prey. Another function is signalling[21]— for instance, a ladybird is less likely to be attacked bypredatory birds that hunt by sight, if it has bold warningcolours, and is also distastefully bitter or poisonous, ormimics other distasteful insects. A young bird may seea warning patterned insect like a ladybird and try to eatit, but it will only do this once; very soon it will spit outthe bitter insect; the other ladybirds in the area will re-main unmolested. The young leopards and ladybirds, in-heriting genes that somehow create spottedness, survive.But while these evolutionary and functional argumentsexplain why these animals need their patterns, they donot explain how the patterns are formed.[69]

• Dirce Beauty butterfly, Colobura dirce

• Grevy’s Zebra, Equus grevyi

• Royal Angelfish, Pygoplites diacanthus

• Leopard, Panthera pardus pardus

• Array of Ladybirds by G.G. Jacobson

• Breeding pattern of Cuttlefish, Sepia officinalis

3.10 Pattern formation

Main article: Pattern formation

Alan Turing,[11] and later the mathematical biologistJames Murray, described a mechanism that sponta-neously creates spotted or striped patterns: a reaction-diffusion system.[70] The cells of a young organism havegenes that can be switched on by a chemical signal, amorphogen, resulting in the growth of a certain type ofstructure, say a darkly pigmented patch of skin. If themorphogen is present everywhere, the result is an evenpigmentation, as in a black leopard. But if it is un-evenly distributed, spots or stripes can result. Turingsuggested that there could be feedback control of theproduction of the morphogen itself. This could causecontinuous fluctuations in the amount of morphogen asit diffused around the body. A second mechanism isneeded to create standing wave patterns (to result inspots or stripes): an inhibitor chemical that switches offproduction of the morphogen, and that itself diffuses

through the body more quickly than the morphogen, re-sulting in an activator-inhibitor scheme. The Belousov–Zhabotinsky reaction is a non-biological example of thiskind of scheme, a chemical oscillator.[70]

Later research has managed to create convincing modelsof patterns as diverse as zebra stripes, giraffe blotches,jaguar spots (medium-dark patches surrounded by darkbroken rings) and ladybird shell patterns (different geo-metrical layouts of spots and stripes, see illustrations).[71]Richard Prum's activation-inhibition models, developedfrom Turing’s work, use six variables to account for theobserved range of nine basic within-feather pigmenta-tion patterns, from the simplest, a central pigment patch,via concentric patches, bars, chevrons, eye spot, pairof central spots, rows of paired spots and an array ofdots.[72][73]:6 More elaborate models simulate complexfeather patterns in the Guinea fowl, Numida meleagris,in which the individual feathers feature transitions frombars at the base to an array of dots at the far (distal) end.These require an oscillation created by two inhibiting sig-nals, with interactions in both space and time.[73]:7–8

Patterns can form for other reasons in the vegetated land-scape of tiger bush[74] and fir waves.[75] Tiger bush stripesoccur on arid slopes where plant growth is limited by rain-fall. Each roughly horizontal stripe of vegetation effec-tively collects the rainwater from the bare zone imme-diately above it.[74] Fir waves occur in forests on moun-tain slopes after wind disturbance, during regeneration.When trees fall, the trees that they had sheltered be-come exposed and are in turn more likely to be dam-aged, so gaps tend to expand downwind. Meanwhile,on the windward side, young trees grow, protected bythe wind shadow of the remaining tall trees.[75] Natu-ral patterns are sometimes formed by animals, as in theMima mounds of the Northwestern United States andsome other areas, which appear to be created over manyyears by the burrowing activities of pocket gophers.[76]

• Giant pufferfish, Tetraodon mbu

• Detail of Giant pufferfish skin pattern

• Snapshot of simulation of Belousov-Zhabotinsky re-action

• Guinea fowl, Numida meleagris feathers transitionfrom barred to spotted, both in-feather and acrossthe bird

• Aerial view of a tiger bush plateau in Niger

9

• Fir waves in White Mountains, New Hampshire

4 See also• Emergence

• Evolutionary history of plants

• Mathematics and art

• Patterned ground

5 References[1] Stevens, Peter. Patterns in Nature, 1974. Page 3.

[2] Balaguer, Mark (12 May 2004, revised 7 April 2009).“Stanford Encyclopedia of Philosophy”. Platonism inMetaphysics. Stanford University. Retrieved 4 May 2012.

[3] The so-called Pythagoreans, who were the first to take upmathematics, not only advanced this subject, but saturatedwith it, they fancied that the principles of mathematicswere the principles of all things. Aristotle, Metaphysics1–5 , cc. 350 BC

[4] Aristotle reports Empedocles arguing that, extquot-edbl[w]herever, then, everything turned out as it wouldhave if it were happening for a purpose, there the crea-tures survived, being accidentally compounded in a suit-able way; but where this did not happen, the creatures per-ished.” The Physics, B8, 198b29 in Kirk, et. al., 304).

[5] Singh, Parmanand. Acharya Hemachandra and the (socalled) Fibonacci Numbers. Math. Ed. Siwan, 20(1):28–30, 1986. ISSN 0047-6269

[6] Knott, Ron. “Fibonacci’s Rabbits”. University of SurreyFaculty of Engineering and Physical Sciences.

[7] About D'Arcy. D' Arcy 150. University of Dundee andtheUniversity of St Andrews. Retrieved 16October 2012.

[8] Stewart, Ian. 2001. Pages 108–109.

[9] Ball, Philip. Shapes. 2009. Page 41.

[10] Hannavy, John (2007). Encyclopedia of Nineteenth-Century Photography 1. CRC Press. p. 149. ISBN 0-415-97235-3.

[11] Turing, A. M. (1952). “The Chemical Basis of Morpho-genesis”. Philosophical Transactions of the Royal Soci-ety B 237 (641): 37–72. Bibcode:1952RSPTB.237...37T.doi:10.1098/rstb.1952.0012.

[12] Ball, Philip. Shapes. 2009. Pages 163, 247–250.

[13] Rozenberg, Grzegorz; Salomaa, Arto. The mathematicaltheory of L systems. Academic Press, New York, 1980.ISBN 0-12-597140-0

[14] Mandelbrot, Benoît B. (1983). The fractal geometry ofnature. Macmillan.

[15] Forbes, Peter. All that useless beauty. The Guardian. Re-view: Non-fiction. 11 February 2012.

[16] Stevens, Peter. 1994. Page 222.

[17] Steen, L.A.. The Science of Patterns. Science, 240: 611–616, 1998. Summary at ascd.org

[18] Devlin, Keith. Mathematics: The Science of Patterns: TheSearch for Order in Life, Mind and the Universe (ScientificAmerican Paperback Library) 1996

[19] Tatarkiewicz, Władysław. Perfection in the Sciences. II.Perfection in Physics and Chemistry, Dialectics and Hu-manism, vol. VII, no. 2 (spring 1980), p. 139.

[20] Darwin, Charles. On the Origin of Species. 1859, chapter4.

[21] Wickler, W. (1968). Mimicry in plants and animals. NewYork: McGraw-Hill.

[22] Poulin, R.; Grutter, A.S. (1996) extquotedblCleaningsymbiosis: proximate and adaptive explanations extquot-edbl. Bioscience 46(7): 512–517. (subscription required)

[23] Koning, Ross (1994). “Plant Physiology InformationWebsite”. Pollination Adaptations. Ross Koning. Re-trieved May 2, 2012.

[24] Stewart, Ian. 2001. Pages 48-49.

[25] Stewart, Ian. 2001. Pages 64-65.

[26] Stewart, Ian. 2001. Page 52.

[27] Stewart, Ian. 2001. Pages 82-84.

[28] Stewart, Ian. 2001. Page 60.

[29] Stewart, Ian. 2001. Page 71.

[30] Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan(2002). “Animal Diversity (Third Edition) extquotedbl.Chapter 8: Acoelomate Bilateral Animals. McGraw-Hill.p. 139. Retrieved October 25, 2012.

[31] Sumrall, Colin D.; Wray, Gregory A. (January 2007).“Ontogeny in the fossil record: diversification of bodyplans and the evolution of “aberrant” symmetry in Pa-leozoic echinoderms”. Paleobiology 33 (1): 149–163.doi:10.1666/06053.1.

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[32] Falconer, Kenneth (2003). Fractal Geometry: Mathemat-ical Foundations and Applications. John Wiley.

[33] Briggs, John (1992). Fractals:The Patterns of Chaos.Thames and Hudson. p. 148.

[34] Meyer, Yves; Roques, Sylvie (1993). Progress in waveletanalysis and applications: proceedings of the InternationalConference “Wavelets and Applications,” Toulouse, France– June 1992. Atlantica Séguier Frontières. p. 25.

[35] Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz,Przemyslaw (2000). Pattern formation in biology, vi-sion and dynamics. World Scientific. p. 78. ISBN9789810237929.

[36] Hahn, Horst K.; Georg,Manfred; Peitgen, Heinz-Otto.(2005). “Fractal aspects of three-dimensional vascularconstructive optimization”. In Losa, Gabriele A.; Non-nenmacher, Theo F. Fractals in biology and medicine.Springer. pp. 55–66.

[37] Addison, Paul S. (1997). Fractals and chaos: an illus-trated course. CRC Press. pp. 44–46.

[38] Maor, Eli. e: The Story of a Number. Princeton Univer-sity Press, 2009. Page 135.

[39] Ball, 2009. Shapes pp 29-32.

[40] “Spiral Lattices & Parastichy”. Smith College. Retrieved24 September 2013.

[41] Kappraff, Jay (2004). “Growth in Plants: A Study inNumber”. Forma 19: 335–354.

[42] Ball, Philip. Shapes. 2009. Page 13.

[43] Coxeter, H. S. M. (1961). Introduction to geometry. Wi-ley. p. 169.

[44] Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid(1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.

[45] Levitov LS (15 March 1991). “Energetic Approachto Phyllotaxis”. Europhys. Lett. 14 (6): 533–9. Bibcode:1991EL.....14..533L. doi:10.1209/0295-5075/14/6/006.(subscription required)

[46] Douady, S; Couder, Y. (March 1992).“Phyllotaxis as a physical self-organized growthprocess”. Physical Review Letters 68 (13):2098–2101. Bibcode:1992PhRvL..68.2098D.doi:10.1103/PhysRevLett.68.2098. PMID10045303.(subscription required)

[47] Ball, Philip. Shapes. 2009. Pages 163, 249–250.

[48] Lorenz, Edward N. (March 1963). “DeterministicNonperiodic Flow”. Journal of the At-mospheric Sciences 20 (2): 130–141.Bibcode:1963JAtS...20..130L. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. ISSN 1520-0469. Retrieved 3 June 2010.

[49] Elaydi, Saber N. (1999). Discrete Chaos. Chapman &Hall/CRC. p. 117.

[50] Ruelle, David. Chance and Chaos. Princeton UniversityPress, 1991.

[51] Wolfram, Stephen. A New Kind of Science. WolframMe-dia, 2002.

[52] von Kármán, Theodore. Aerodynamics. McGraw-Hill(1963): ISBN 978-0070676022. Dover (1994): ISBN978-0486434858.

[53] Lewalle, Jacques (2006). “Flow Separation and Sec-ondary Flow: Section 9.1”. Lecture Notes in Incompress-ible Fluid Dynamics: Phenomenology, Concepts and Ana-lytical Tools. Syracuse, NY: Syracuse University..

[54] French, A.P. Vibrations and Waves. Nelson Thornes,1971.

[55] Tolman, H.L. (2008), “Practical wind wave modeling”, inMahmood, M.F., CBMS Conference Proceedings on Wa-ter Waves: Theory and Experiment, Howard University,USA, 13–18 May 2008: World Scientific Publ.

[56] “Types of Dunes”. USGS. 29 October 1997. RetrievedMay 2, 2012.

[57] Strahler, A. & Archibold, O.W. Physical Geography: Sci-ence and Systems of the Human Environment. JohnWiley,4th edition 2008. Page 442.

[58] Schwämmle, V.; Herrman, H.J. (2003). “Solitarywave behaviour of sand dunes”. Nature 426 (Dec.11): 619–620 Abstract. Bibcode:2003Natur.426..619S.doi:10.1038/426619a. PMID 14668849.(subscription re-quired)

[59] Philip Ball. Shapes, 2009. Page 68.

[60] Frederick J. Almgren, Jr. and Jean E. Taylor, The geom-etry of soap films and soap bubbles, Scientific American,vol. 235, pp. 82–93, July 1976.

[61] Ball, Philip. Shapes, 2009. Pages 73–76.

[62] Ball, Philip. Shapes. 2009. Pages 96–101.

[63] Brodie, Christina (February 2005). “Geometry and Pat-tern in Nature 3: The holes in radiolarian and diatomtests”. Microscopy-UK. Retrieved May 28, 2012.

6.2 General books 11

[64] Ball, Philip. Shapes. 2009. Pages 51–54.

[65] Armstrong, M.A. (1988). Groups and Symmetry. NewYork: Springer-Verlag.

[66] Hook, J.R.; Hall, H.E. Solid State Physics (2nd Edition).Manchester Physics Series, John Wiley & Sons, 2010.ISBN 978-0-471-92804-1

[67] Stevens, Peter. 1974. Page 207.

[68] Stevens, Peter. 1974. Page 208.

[69] Ball, Philip. Shapes. 2009. Pages 156–158.

[70] Ball, Philip. Shapes. 2009. Pages 159–167.

[71] Ball, Philip. Shapes. 2009. Pages 168–180.

[72] Rothenburg, David. 2011. Pages 93–95.

[73] Prum, Richard O.; Williamson, Scott (2002). “Reaction–diffusion models of within-feather pigmentation pattern-ing”. Proceedings Royal Society London B 269: 781–792.doi:10.1098/rspb.2001.1896.

[74] Tongway, D.J., Valentin, C. & Seghieri, J. (2001).Banded vegetation patterning in arid and semiarid envi-ronments. New York: Springer-Verlag.

[75] D'Avanzo, C. (22 February 2004). “Fir Waves: Regener-ation in New England Conifer Forests”. TIEE. Retrieved26 May 2012.

[76] Morelle, Rebecca. extquotedbl‘Digital gophers’ solveMima mound mystery”. BBC News. Retrieved 9 Decem-ber 2013.

6 Bibliography

6.1 Pioneering authors

• Fibonacci, Leonardo. Liber Abaci, 1202.

--- ----- translated by Sigler, Laurence E. Fi-bonacci’s Liber Abaci. Springer, 2002.

• Haeckel, Ernst. Kunstformen der Natur (Art Formsin Nature), 1899–1904.

• Thompson, D'Arcy Wentworth. On Growth andForm. Cambridge, 1917.

6.2 General books

• Adam, John A. Mathematics in Nature: ModelingPatterns in the Natural World. Princeton UniversityPress, 2006.

• Ball, Philip. Nature’s Patterns: a tapestry in threeparts. 1:Shapes. 2:Flow. 3:Branches. Oxford, 2009.

• Murphy, Pat and Neill, William. By Nature’s De-sign. Chronicle Books, 1993.

• Rothenburg, David. Survival of the Beautiful: Art,Science and Evolution. Bloomsbury Press, 2011.

• Stevens, Peter S. Patterns in Nature. Little, Brown& Co, 1974.

• Stewart, Ian. What Shape is a Snowflake? MagicalNumbers in Nature. Weidenfeld & Nicolson, 2001.

6.3 Patterns from nature (as art)

• Edmaier, Bernard. Patterns of the Earth. PhaidonPress, 2007.

• Macnab, Maggie. Design by Nature: Using Univer-sal Forms and Principles in Design. New Riders,2012.

• Nakamura, Shigeki. Pattern Sourcebook: 250 Pat-terns Inspired by Nature.. Books 1 and 2. Rockport,2009.

• O'Neill, Polly. Surfaces and Textures: A VisualSourcebook. Black, 2008.

• Porter, Eliot, and Gleick, James. Nature’s Chaos.Viking Penguin, 1990.

7 External links• Fibonacci Numbers and the Golden Section

• Phyllotaxis: an Interactive Site for theMathematicalStudy of Plant Pattern Formation

12 8 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

8 Text and image sources, contributors, and licenses

8.1 Text• Patterns in nature Source: http://en.wikipedia.org/wiki/Patterns_in_nature?oldid=626975484 Contributors: Alan Liefting, Viriditas,

BD2412, Bruce1ee, Sonitus, Wavelength, Gaius Cornelius, Tamfang, Fmindlin, Cybercobra, Valenciano, Novangelis, Neoking, Mirror-mundo, Tillman, Lopkiol, David Eppstein, Ampy1, Chiswick Chap, DadaNeem, Paradoctor, Yintan, ImageRemovalBot, Elassint, Piled-higheranddeeper, CharlieRCD, Excirial, MrOllie, Jarble, Decora, 3family6, Omnipaedista, Edgars2007, FrescoBot, Patchy1, Pinethicket,Tomcat7, Animalparty, GA bot, John Cline, Medeis, ClueBot NG, Gilderien, Mesoderm, Widr, KLBot2, Bibcode Bot, Kempf EK,M0rphzone, Lucquessoy, Dariusz wozniak, Spectral sequence, Saleh Masoumi, Faroffthunder, Anrnusna, Monkbot and Anonymous: 16

8.2 Images• File:Aphids_and_live_young_under_Sycamore_leaf.JPG Source: http://upload.wikimedia.org/wikipedia/commons/d/d1/Aphids_

and_live_young_under_Sycamore_leaf.JPG License: CC-BY-SA-3.0 Contributors: Own work Original artist: Chiswick Chap• File:Cycas_circinalis_male_cone_in_Olomouc.jpg Source: http://upload.wikimedia.org/wikipedia/commons/e/ee/Cycas_circinalis_

male_cone_in_Olomouc.jpg License: CC-BY-2.5 Contributors: Own work Original artist: Michal Maňas (User:snek01)• File:Dragon_trees.jpg Source: http://upload.wikimedia.org/wikipedia/commons/7/74/Dragon_trees.jpg License: Public domain Con-

tributors: ? Original artist: ?• File:Drcy.svg Source: http://upload.wikimedia.org/wikipedia/commons/2/24/Drcy.svg License: Public domain Contributors: D'Arcy

Thompson Original artist: D'Arcy Thompson• File:Symbol_support_vote.svg Source: http://upload.wikimedia.org/wikipedia/en/9/94/Symbol_support_vote.svg License: ? Contribu-

tors: ? Original artist: ?• File:The_Desert_Ultra_-_Sand_Dunes.jpg Source: http://upload.wikimedia.org/wikipedia/commons/f/f6/The_Desert_Ultra_-_Sand_

Dunes.jpg License: CC-BY-SA-3.0 Contributors: Own work Original artist: Worldwide Happy Media• File:Wikisource-logo.svg Source: http://upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg License: ? Contributors:

Original artist: Nicholas Moreau• File:Yemen_Chameleon_(cropped).jpg Source: http://upload.wikimedia.org/wikipedia/commons/f/fc/Yemen_Chameleon_

%28cropped%29.jpg License: CC-BY-SA-3.0 Contributors: Own work Original artist: Chiswick Chap

8.3 Content license• Creative Commons Attribution-Share Alike 3.0