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Patterns in nature are visible regularities of form found in the
natural world ''Natural World'' is a strand of British wildlife documentary programmes broadcast on BBC Two and BBC Two HD and regarded by the BBC as its flagship natural history series. It is the longest-running documentary in its genre on British televis ...
. These
patterns A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...
recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries,
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
,
spirals In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:meanders,
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s,
foams Foams are materials formed by trapping pockets of gas in a liquid or solid. A bath sponge and the head on a glass of beer are examples of foams. In most foams, the volume of gas is large, with thin films of liquid or solid separating the ...
,
tessellations A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
, cracks and stripes. Early
Greek philosophers Ancient Greek philosophy arose in the 6th century BC, marking the end of the Greek Dark Ages. Greek philosophy continued throughout the Hellenistic period and the period in which Greece and most Greek-inhabited lands were part of the Roman Empire ...
studied pattern, with
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
,
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
and
Empedocles Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
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 Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s, leading him to formulate the concept of a
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist
D'Arcy Thompson Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology, travelled on expeditions to the Bering Strait a ...
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 Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
predicted mechanisms of
morphogenesis Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of deve ...
which give rise to
patterns A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated li ...
of spots and stripes. The Hungarian biologist
Aristid Lindenmayer Aristid Lindenmayer (17 November 1925 – 30 October 1989) was a Hungarian biologist. In 1968 he developed a type of formal languages that is today called L-systems or Lindenmayer Systems. Using those systems Lindenmayer modelled the behaviour ...
and the French American mathematician
Benoît Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
showed how the mathematics of
fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
could create plant growth patterns. Mathematics,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and chemistry can explain patterns in nature at different levels and scales. Patterns in living things are explained by the biological processes of
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...
and
sexual selection Sexual selection is a mode of natural selection in which members of one biological sex choose mates of the other sex to mate with (intersexual selection), and compete with members of the same sex for access to members of the opposite sex ( ...
. Studies of
pattern formation The science of pattern formation deals with the visible, ( statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature. In developmental biology, pattern formation refers to the generation of ...
make use of
computer models Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...
to simulate a wide range of patterns.


History

Early Greek philosophers attempted to explain order in
nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
, anticipating modern concepts.
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
(c. 570–c. 495 BC) explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.
Empedocles Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
(c. 494–c. 434 BC) to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(c. 427–c. 347 BC) argued for the existence of natural universals. He considered these to consist of ideal forms ( ''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.
Theophrastus Theophrastus (; grc-gre, Θεόφραστος ; c. 371c. 287 BC), a Greek philosopher and the successor to Aristotle in the Peripatetic school. He was a native of Eresos in Lesbos.Gavin Hardy and Laurence Totelin, ''Ancient Botany'', Routle ...
(c. 372–c. 287 BC) noted that plants "that have flat leaves have them in a regular series";
Pliny the Elder Gaius Plinius Secundus (AD 23/2479), called Pliny the Elder (), was a Roman author, naturalist and natural philosopher, and naval and army commander of the early Roman Empire, and a friend of the emperor Vespasian. He wrote the encyclopedic ' ...
(23–79 AD) noted their patterned circular arrangement. Centuries later,
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
(1452–1519) noted the spiral arrangement of leaf patterns, that tree trunks gain successive rings as they age, and proposed a rule purportedly satisfied by the cross-sectional areas of tree-branches. In 1202,
Leonardo Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
introduced the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
to the western world with his book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
''. Fibonacci presented a
thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
on the growth of an idealized rabbit population. Johannes Kepler (1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the pentagonal form of some flowers. 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 Platonic realism is the philosophical position that universals or abstract objects exist objectively and outside of human minds. It is named after the Greek philosopher Plato who applied realism to such universals, which he considered ideal for ...
of the
quincunx A quincunx () is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center. The same pattern has other names, including "in saltire" or "in cross" in heraldry (d ...
pattern. The discourse's central chapter features examples and observations of the quincunx in botany. In 1754, Charles Bonnet observed that the spiral
phyllotaxis In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic arrangements of leaves on a stem are opposite and alterna ...
of plants were frequently expressed in both
clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...
and counter-clockwise
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
series. Mathematical observations of phyllotaxis followed with Karl Friedrich Schimper and his friend
Alexander Braun Alexander Carl Heinrich Braun (10 May 1805 – 29 March 1877) was a German botanist from Regensburg, Bavaria. His research centered on the morphology of plants. Biography He studied botany in Heidelberg, Paris and Munich. In 1833 he began teachi ...
's 1830 and 1830 work, respectively;
Auguste Bravais Auguste Bravais (; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied ...
and his brother Louis connected phyllotaxis ratios to the Fibonacci sequence in 1837, also noting its appearance in
pinecone A conifer cone (in formal botanical usage: strobilus, plural strobili) is a seed-bearing organ on gymnosperm plants. It is usually woody, ovoid to globular, including scales and bracts arranged around a central axis, especially in conifers an ...
s and
pineapple The pineapple (''Ananas comosus'') is a tropical plant with an edible fruit; it is the most economically significant plant in the family Bromeliaceae. The pineapple is indigenous to South America, where it has been cultivated for many centuri ...
s. In his 1854 book, German psychologist Adolf Zeising explored the golden ratio expressed in the arrangement of plant parts, the skeletons of animals and the branching patterns of their veins and nerves, as well as in
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s. In the 19th century, the Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
with a given boundary, which is now named after him. He studied soap films intensively, formulating
Plateau's laws Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws. Laws ...
which describe the structures formed by films in foams.
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
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 In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
; the
Beijing National Aquatics Center The National Aquatics Centre (), and colloquially known as the Water Cube () and the Ice Cube (), is an aquatics center at the Olympic Green in Beijing, China. The facility was originally constructed to host the aquatics competitions at the 2 ...
adapted the structure for their outer wall in the 2008 Summer Olympics. Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular
Radiolaria The Radiolaria, also called Radiozoa, are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm. The el ...
, emphasising their symmetry to support his faux-
Darwinian Darwinism is a theory of biological evolution developed by the English naturalist Charles Darwin (1809–1882) and others, stating that all species of organisms arise and develop through the natural selection of small, inherited variations that ...
theories of evolution. The American photographer
Wilson Bentley Wilson Alwyn Bentley (February 9, 1865 – December 23, 1931), also known as Snowflake Bentley, was an American meteorologist and photographer, who was the first known person to take detailed photographs of snowflakes and record their featu ...
took the first micrograph of a
snowflake A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, which falls through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. ...
in 1885. In the 20th century, A. H. Church studied the patterns of phyllotaxis in his 1904 book. In 1917,
D'Arcy Wentworth Thompson Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology Mathematical and theoretical biology, or biomat ...
published ''
On Growth and Form ''On Growth and Form'' is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many top ...
''; 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 Animals are multicellular, eukaryotic organisms in the Kingdom (biology), biological kingdom Animalia. With few exceptions, animals Heterotroph, consume organic material, Cellular respiration#Aerobic respiration, breathe oxygen, are Motilit ...
s and
mollusc shell The mollusc (or molluskOften spelled mollusk shell in the USA; the spelling "mollusc" are preferred by ) shell is typically a calcareous exoskeleton which encloses, supports and protects the soft parts of an animal in the phylum Mollusca, wh ...
s. In 1952, the computer scientist
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
(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 Morphogenesis (from the Greek ''morphê'' shape and ''genesis'' creation, literally "the generation of form") is the biological process that causes a cell, tissue or organism to develop its shape. It is one of three fundamental aspects of deve ...
. He predicted oscillating
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...
s, in particular the
Belousov–Zhabotinsky reaction A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
. These activator-inhibitor mechanisms can, Turing suggested, generate patterns (dubbed " Turing patterns") of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis. In 1968, the Hungarian theoretical biologist
Aristid Lindenmayer Aristid Lindenmayer (17 November 1925 – 30 October 1989) was a Hungarian biologist. In 1968 he developed a type of formal languages that is today called L-systems or Lindenmayer Systems. Using those systems Lindenmayer modelled the behaviour ...
(1925–1989) developed the
L-system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...
, a
formal grammar In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...
which can be used to model plant growth patterns in the style of fractals. Rozenberg, Grzegorz; Salomaa, Arto. ''The Mathematical Theory of L Systems''.
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes referen ...
, New York, 1980. L-systems have an
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
of symbols that can be combined using 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 Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
,
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, Helge von Koch, Wacław Sierpiński and others,
Benoît Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
wrote a famous paper, ''
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that ...
'', crystallising mathematical thought into the concept of the fractal. File:Cycas circinalis male cone in Olomouc.jpg,
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
patterns occur widely in plants such as this queen sago, ''
Cycas circinalis ''Cycas circinalis'', also known as the queen sago, is a species of cycad known in the wild only from southern India. ''Cycas circinalis'' is the only gymnosperm species found among native Sri Lankan flora. Taxonomy ''C. circinallis'' is native ...
''. File:National Aquatics Center Construction (cropped).jpg, Beijing's National Aquatics Center for the 2008 Olympic games has a
Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
. File:Drcy.svg, D'Arcy Thompson pioneered the study of growth and form in his 1917 book.


Causes

Living things like
orchid Orchids are plants that belong to the family Orchidaceae (), a diverse and widespread group of flowering plants with blooms that are often colourful and fragrant. Along with the Asteraceae, they are one of the two largest families of flowerin ...
s, hummingbirds, and the peacock's tail have abstract designs with a beauty of form, pattern and colour that artists struggle to match.Forbes, Peter. ''All that useless beauty''. The Guardian. Review: Non-fiction. 11 February 2012. 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. Mathematics seeks to discover and explain abstract patterns or regularities of all kinds. Devlin, Keith. ''Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe'' (Scientific American Paperback Library) 1996 Visual patterns in nature find explanations in chaos theory, fractals, logarithmic spirals, topology and other mathematical patterns. For example,
L-system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...
s form convincing models of different patterns of tree growth. The laws of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
apply the abstractions of mathematics to the real world, often as if it were perfect. For example, a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
is perfect when it has no structural defects such as dislocations and is fully symmetric. Exact mathematical perfection can only approximate real objects. Visible patterns in nature are governed by
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
s; for example,
meander A meander is one of a series of regular sinuous curves in the channel of a river or other watercourse. It is produced as a watercourse erodes the sediments of an outer, concave bank ( cut bank) and deposits sediments on an inner, convex ba ...
s can be explained using fluid dynamics. In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...
can cause the development of patterns in living things for several reasons, including
camouflage Camouflage is the use of any combination of materials, coloration, or illumination for concealment, either by making animals or objects hard to see, or by disguising them as something else. Examples include the leopard's spotted coat, the b ...
, Darwin, Charles. ''On the Origin of Species''. 1859, chapter 4.
sexual selection Sexual selection is a mode of natural selection in which members of one biological sex choose mates of the other sex to mate with (intersexual selection), and compete with members of the same sex for access to members of the opposite sex ( ...
, and different kinds of signalling, including mimicry and cleaning symbiosis. In plants, the shapes, colours, and patterns of
insect-pollinated Entomophily or insect pollination is a form of pollination whereby pollen of plants, especially but not only of flowering plants, is distributed by insects. Flowers pollinated by insects typically advertise themselves with bright colours, som ...
flowers A flower, sometimes known as a bloom or blossom, is the reproductive structure found in flowering plants (plants of the division Angiospermae). The biological function of a flower is to facilitate reproduction, usually by providing a mechanism ...
like the
lily ''Lilium'' () is a genus of herbaceous flowering plants growing from bulbs, all with large prominent flowers. They are the true lilies. Lilies are a group of flowering plants which are important in culture and literature in much of the world. M ...
have evolved to attract insects such as bees. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as nectar guides that can be seen at a distance.


Types of pattern


Symmetry

Symmetry is pervasive in living things. Animals mainly have bilateral or
mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
, as do the leaves of plants and some flowers such as
orchid Orchids are plants that belong to the family Orchidaceae (), a diverse and widespread group of flowering plants with blooms that are often colourful and fragrant. Along with the Asteraceae, they are one of the two largest families of flowerin ...
s. Plants often have radial or
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. Among non-living things,
snowflake A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, which falls through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. ...
s have striking 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.
Crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s in general have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike
quasicrystals A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical ...
). Rotational symmetry is found at different scales among non-living things, including the crown-shaped splash pattern formed when a drop falls into a pond, and both the spheroidal shape and rings of a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
like Saturn. 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). 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. File:Tiger-berlin-5 symmetry.jpg, Animals often show mirror or bilateral symmetry, like this
tiger The tiger (''Panthera tigris'') is the largest living cat species and a member of the genus ''Panthera''. It is most recognisable for its dark vertical stripes on orange fur with a white underside. An apex predator, it primarily preys on ...
. File:Starfish 02 (paulshaffner) cropped.jpg, Echinoderms like this starfish have fivefold symmetry. File:Medlar 5-symmetry.jpg, Fivefold symmetry can be seen in many flowers and some fruits like this
medlar ''Mespilus germanica'', known as the medlar or common medlar, is a large shrub or small tree in the rose family Rosaceae. The fruit of this tree, also called medlar, has been cultivated since Roman times, is usually available in winter and ea ...
. File:Schnee2.jpg,
Snowflake A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, which falls through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. ...
s have sixfold symmetry. File:Aragonite-Fluorite-cflu02c.jpg,
Fluorite Fluorite (also called fluorspar) is the mineral form of calcium fluoride, CaF2. It belongs to the halide minerals. It crystallizes in isometric cubic habit, although octahedral and more complex isometric forms are not uncommon. The Mohs sca ...
showing cubic crystal habit. File:Water splashes 001.jpg, Water splash approximates
radial symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pla ...
. File:GarnetCrystalUSGOV.jpg,
Garnet Garnets () are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. All species of garnets possess similar physical properties and crystal forms, but differ in chemical composition. The different s ...
showing rhombic dodecahedral crystal habit. File:Mikrofoto.de-volvox-8.jpg, ''
Volvox ''Volvox'' is a polyphyletic genus of chlorophyte green algae in the family Volvocaceae. It forms spherical colonies of up to 50,000 cells. They live in a variety of freshwater habitats, and were first reported by Antonie van Leeuwenhoek in 170 ...
'' has spherical symmetry. File:Two Oceans Aquarium03.jpg, Sea anemones have
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
.


Trees, fractals

The branching pattern of trees was described in the
Italian Renaissance The Italian Renaissance ( it, Rinascimento ) was a period in Italian history covering the 15th and 16th centuries. The period is known for the initial development of the broader Renaissance culture that spread across Europe and marked the trans ...
by
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
. In '' A Treatise on Painting'' he stated that:
All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk elow them
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. 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
right-angled triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
. One explanation is that this allows trees to better withstand high winds. Simulations of biomechanical models agree with the rule. Fractals are infinitely self-similar, iterated mathematical constructs having
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
. Infinite
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of
fern A fern (Polypodiopsida or Polypodiophyta ) is a member of a group of vascular plants (plants with xylem and phloem) that reproduce via spores and have neither seeds nor flowers. The polypodiophytes include all living pteridophytes exce ...
s and
umbellifer Apiaceae or Umbelliferae is a family of mostly aromatic flowering plants named after the type genus '' Apium'' and commonly known as the celery, carrot or parsley family, or simply as umbellifers. It is the 16th-largest family of flowering plant ...
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 Bryozoa (also known as the Polyzoa, Ectoprocta or commonly as moss animals) are a phylum of simple, aquatic invertebrate animals, nearly all living in sedentary colonies. Typically about long, they have a special feeding structure called a ...
,
coral Corals are marine invertebrates within the class Anthozoa of the phylum Cnidaria. They typically form compact colonies of many identical individual polyps. Coral species include the important reef builders that inhabit tropical oceans and ...
s,
hydrozoa Hydrozoa (hydrozoans; ) are a taxonomic class of individually very small, predatory animals, some solitary and some colonial, most of which inhabit saline water. The colonies of the colonial species can be large, and in some cases the specialize ...
like the
air fern The air fern (''Sertularia argentea'') is a species of marine animal in the family Sertulariidae. It is also known as the sea fir and Neptune plant. These so-called "ferns" are dead and dried colonies of hydrozoans, colonies of marine hydroid ...
, ''Sertularia argentea'', and in non-living things, notably electrical discharges.
Lindenmayer system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into so ...
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 length), and number of branches per branch point. Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic
fault line In geology, a fault is a planar fracture or discontinuity in a volume of rock across which there has been significant displacement as a result of rock-mass movements. Large faults within Earth's crust result from the action of plate tectonic ...
s,
mountain A mountain is an elevated portion of the Earth's crust, generally with steep sides that show significant exposed bedrock. Although definitions vary, a mountain may differ from a plateau in having a limited summit area, and is usually highe ...
s,
coastline The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in ...
s, animal coloration, snow flakes,
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s,
blood vessel The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away ...
branching,
Purkinje cells Purkinje cells, or Purkinje neurons, are a class of GABAergic inhibitory neurons located in the cerebellum. They are named after their discoverer, Czech anatomist Jan Evangelista Purkyně, who characterized the cells in 1839. Structure T ...
, actin cytoskeletons, and ocean waves. File:Dragon trees.jpg, The growth patterns of certain trees resemble these
Lindenmayer system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into so ...
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 Pinnation (also called pennation) is the arrangement of feather-like or multi-divided features arising from both sides of a common axis. Pinnation occurs in biological morphology, in crystals, such as some forms of ice or metal crystals, and in ...
, not infinite File:Romanesco broccoli (Brassica oleracea).jpg, Fractal spirals: Romanesco broccoli showing self-similar form File:Angelica flowerhead showing pattern.JPG,
Angelica ''Angelica'' is a genus of about 60 species of tall biennial and perennial herbs in the family Apiaceae, native to temperate and subarctic regions of the Northern Hemisphere, reaching as far north as Iceland, Lapland, and Greenland. They gr ...
flowerhead, a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
made of spheres (self-similar) File:Square1.jpg, Trees: Lichtenberg figure: high voltage
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
breakdown in an acrylic polymer block File:Dendritic Copper Crystals - 20x magnification.jpg, Trees: dendritic copper crystals (in microscope)


Spirals

Spirals In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:molluscs Mollusca is the second-largest phylum of invertebrate animals after the Arthropoda, the members of which are known as molluscs or mollusks (). Around 85,000 extant species of molluscs are recognized. The number of fossil species is estim ...
. For example, in the
nautilus The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina. It comprises six living species in ...
, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
. Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity. Plant spirals can be seen in
phyllotaxis In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic arrangements of leaves on a stem are opposite and alterna ...
, the arrangement of leaves on a stem, and in the arrangement (
parastichy Parastichy, in phyllotaxy, is the spiral pattern of particular plant organs on some plants, such as areoles on cacti stems, florets in sunflower heads and scales in pine cones. These spirals involve the insertion of a single primordium. See al ...
) of other parts as in
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
flower heads and seed heads like the sunflower or
fruit In botany, a fruit is the seed-bearing structure in flowering plants that is formed from the ovary after flowering. Fruits are the means by which flowering plants (also known as angiosperms) disseminate their seeds. Edible fruits in particu ...
structures like the
pineapple The pineapple (''Ananas comosus'') is a tropical plant with an edible fruit; it is the most economically significant plant in the family Bromeliaceae. The pineapple is indigenous to South America, where it has been cultivated for many centuri ...
and 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. Phyllotaxis spirals can be generated from 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 Pears are fruits produced and consumed around the world, growing on a tree and harvested in the Northern Hemisphere in late summer into October. The pear tree and shrub are a species of genus ''Pyrus'' , in the family Rosaceae, bearing the po ...
it is 3/8; in almond it is 5/13. In disc phyllotaxis as in the sunflower and daisy, the florets are arranged along
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...
, but this is disguised because successive florets are spaced far apart, by the golden angle, 137.508° (dividing the circle in the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
); when the flowerhead is mature so all the elements are the same size, this spacing creates a Fibonacci number of more obvious spirals. From the point of view of physics, spirals are lowest-energy configurations which emerge spontaneously through
self-organizing Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order and disorder, order arises from local interactions between parts of an initially disordered system. The process can be spon ...
processes in
dynamic system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
s. 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 Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
s that manipulate the concentration of the plant hormone auxin, which activates
meristem The meristem is a type of tissue found in plants. It consists of undifferentiated cells (meristematic cells) capable of cell division. Cells in the meristem can develop into all the other tissues and organs that occur in plants. These cells conti ...
growth, alongside other mechanisms to control the relative angle of buds around the stem. 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 Photosynthesis is a process used by plants and other organisms to convert light energy into chemical energy that, through cellular respiration, can later be released to fuel the organism's activities. Some of this chemical energy is stored i ...
. File:Fibonacci spiral 34.svg,
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
spiral File:Ovis canadensis 2 (cropped).jpg, Bighorn sheep, ''Ovis canadensis'' File:Aloe polyphylla spiral.jpg, Spirals:
phyllotaxis In botany, phyllotaxis () or phyllotaxy is the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Leaf arrangement The basic arrangements of leaves on a stem are opposite and alterna ...
of spiral aloe, '' Aloe polyphylla'' File:NautilusCutawayLogarithmicSpiral.jpg, ''
Nautilus The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina. It comprises six living species in ...
'' shell's
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
ic growth spiral File:Pflanze-Sonnenblume1-Asio (cropped).JPG,
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...
: 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 spirals


Chaos, flow, meanders

In mathematics, a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
is chaotic if it is (highly) sensitive to initial conditions (the so-called " butterfly effect"), which requires the mathematical properties of topological mixing and
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
periodic orbits In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynami ...
. 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 the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
'' in chaotic systems have a
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
. Some
cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably
Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
's
Rule 30 Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983. Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour. This rule is of particular interest because it pr ...
. Vortex streets are zigzagging patterns of whirling
vortices In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects. Smooth ( 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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
of the fluid.
Meander A meander is one of a series of regular sinuous curves in the channel of a river or other watercourse. It is produced as a watercourse erodes the sediments of an outer, concave bank ( cut bank) and deposits sediments on an inner, convex ba ...
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 Helicoidal flow is the cork-screw-like flow of water in a meander. It is one example of a secondary flow. Helicoidal flow is a contributing factor to the formation of slip-off slopes and river cliffs in a meandering section of the river. The he ...
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 Erosion is the action of surface processes (such as water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location where it is deposited. Erosion is dis ...
accelerates, further increasing the meandering in a powerful
positive feedback loop Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the ...
. File:Textile cone (cropped).JPG, Chaos: shell of gastropod mollusc the cloth of gold cone, '' Conus textile'', resembles
Rule 30 Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983. Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour. This rule is of particular interest because it pr ...
cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tesse ...
File:Vortex-street-1.jpg, Flow: vortex street of clouds at Juan Fernandez Islands File:Rio Negro meanders.JPG, Meanders: dramatic meander scars and oxbow lakes 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''


Waves, dunes

Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it Oscillation, oscillate as they pass by. Wind waves are sea surface waves that create the characteristic chaotic pattern of any large body of water, though their statistical behaviour can be predicted with wind wave models. As waves in water or wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create dunes, 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. Barchans 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 avalanches, 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. Apart from this nonlinearity, barchans behave rather like soliton, solitary waves. 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 dislocations in Sistan, Afghanistan


Bubbles, foam

A soap bubble forms a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, a surface with minimal area (
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
) — 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. A foam is a mass of bubbles; foams of different materials occur in nature. Foams composed of
soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s obey
Plateau's laws Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws. 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). Structures with minimal surfaces can be used as tents. At the scale of living cell (biology), cells, foam patterns are common; radiolarians, sponge spicule (sponge), spicules, silicoflagellate exoskeletons and the calcite skeleton of a sea urchin, ''Cidaris rugosa'', all resemble mineral casts of Plateau foam boundaries. 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). File:Foam - big.jpg, Foam of soap bubbles: 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 The Radiolaria, also called Radiozoa, are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm. The el ...
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 C60: 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.


Tessellations

Tessellations are patterns formed by repeating tiles all over a flat surface. There are 17 wallpaper groups of tilings. 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 wasps, 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 A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
, 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.Hook, J. R.; Hall, H. E. ''Solid State Physics'' (2nd Edition). Manchester Physics Series, John Wiley & Sons, 2010. 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


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. 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. 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)


Spots, stripes

Leopards and ladybirds are spotted; angelfish and zebras are striped. These patterns have an evolutionary 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 Camouflage is the use of any combination of materials, coloration, or illumination for concealment, either by making animals or objects hard to see, or by disguising them as something else. Examples include the leopard's spotted coat, the b ...
; for instance, a leopard that is harder to see catches more prey. Another function is Signalling theory, signalling — for instance, a ladybird is less likely to be attacked by predatory 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 genes 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. 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 ladybirds by Georgij Georgiewitsch Jacobson, G.G. Jacobson File:Sepia officinalis Cuttlefish striped breeding pattern.jpg, Breeding pattern of cuttlefish, ''Sepia officinalis''


Pattern formation

Alan Turing, and later the mathematical biologist James D. Murray, James Murray, described a mechanism that spontaneously creates spotted or striped patterns: a reaction–diffusion system. 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 A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
is a non-biological example of this kind of scheme, a chemical oscillator. 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). 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. 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. Patterns can form for other reasons in the patterned vegetation, vegetated landscape of tiger bush and fir waves. 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. 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. 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 gophers, while the so-called fairy circles of Namibia appear to be created by the interaction of competing groups of sand termites, along with competition for water among the desert plants. 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. 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. 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 A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in ...
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 circles 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


See also

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


References

Footnotes Citations


Bibliography

Pioneering authors * Fibonacci, Fibonacci, Leonardo. ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describe ...
'', 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 ''On Growth and Form'' is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many top ...
''. Cambridge, 1917. General books * Adam, John A
''Mathematics in Nature: Modeling Patterns in the Natural World''
Princeton University Press, 2006. * * * * Ball, Philip. ''Patterns in Nature''. Chicago, 2016. * Pat Murphy (writer), Murphy, Pat and Neill, William. ''By Nature's Design''. Chronicle Books, 1993. * * * 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


Fibonacci Numbers and the Golden Section

Phyllotaxis: an Interactive Site for the Mathematical Study of Plant Pattern Formation
{{Authority control Applied mathematics History of science Nature Pattern formation Patterns Recreational mathematics