Patterns In Nature
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Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be
modelled mathematically A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics ...
. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Early Greek philosophers 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 institutio ...
,
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 politic ...
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 Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repea ...
examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundreds of
marine organisms Marine life, sea life, or ocean life is the plants, animals and other organisms that live in the salt water of seas or oceans, or the brackish water of coastal estuaries. At a fundamental level, marine life affects the nature of the planet. M ...
to emphasise their
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician
Alan Turing 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 c ...
predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.
Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 rel ...
and chemistry can explain patterns in nature at different levels and scales. Patterns in living things are explained by the
biological 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 ...
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. Studies of pattern formation make use of computer models to simulate a wide range of patterns.


History

Early Greek philosophers attempted to explain order in
nature 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 ar ...
, 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 politic ...
(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 Darwin may refer to: Common meanings * Charles Darwin (1809–1882), English naturalist and writer, best known as the originator of the theory of biological evolution by natural selection * Darwin, Northern Territory, a territorial capital city i ...
'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 institutio ...
(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 (c. 372–c. 287 BC) noted that plants "that have flat leaves have them in a regular series"; Pliny the Elder (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 re ...
(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 introduced the Fibonacci sequence to the western world with his book '' Liber Abaci''. 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 anci ...
on the growth of an idealized
rabbit Rabbits, also known as bunnies or bunny rabbits, are small mammals in the family Leporidae (which also contains the hares) of the order Lagomorpha (which also contains the pikas). ''Oryctolagus cuniculus'' includes the European rabbit speci ...
population. Johannes Kepler (1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the
pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...
al form of some flowers. In 1658, the English physician and philosopher
Sir Thomas Browne Sir Thomas Browne (; 19 October 1605 – 19 October 1682) was an English polymath and author of varied works which reveal his wide learning in diverse fields including science and medicine, religion and the esoteric. His writings display a ...
discussed "how Nature Geometrizeth" in ''
The Garden of Cyrus ''The Garden of Cyrus'', or ''The Quincuncial Lozenge, or Network Plantations of the Ancients, naturally, artificially, mystically considered'', is a discourse by Sir Thomas Browne. First published in 1658, along with its diptych companion '' Ur ...
'', citing
Pythagorean numerology Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in ...
involving the number 5, and the Platonic form 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 (de ...
pattern. The discourse's central chapter features examples and observations of the quincunx in botany. In 1754, Charles Bonnet observed that the spiral phyllotaxis of plants were frequently expressed in both clockwise 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 Karl Friedrich Schimper (15 February 1803 – 21 December 1867) was a German botany, botanist, Natural science, naturalist and poet. Life Early life and education Schimper was born in Mannheim, on February 15, 1803, to Friedrich Ludwig ...
and his friend Alexander Braun's 1830 and 1830 work, respectively; Auguste Bravais and his brother Louis connected phyllotaxis ratios to the Fibonacci sequence in 1837, also noting its appearance in pinecones and pineapples. In his 1854 book, German psychologist Adolf Zeising explored the golden ratio expressed in the arrangement of plant parts, the
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
s 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, macr ...
s. In the 19th century, the Belgian physicist
Joseph Plateau Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repea ...
(1801–1883) formulated the
mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...
of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams. Lord Kelvin identified the problem of the most efficient way to pack cells of equal volume as a foam in 1887; his solution uses just one solid, the bitruncated cubic honeycomb with very slightly curved faces to meet Plateau's laws. No better solution was found until 1993 when Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure; the Beijing National Aquatics Center adapted the structure for their outer wall in the 2008 Summer Olympics. Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
to support his faux- Darwinian theories of evolution. The American photographer Wilson Bentley took the first micrograph of a snowflake in 1885. In the 20th century,
A. H. Church A is the first letter of the Latin and English alphabet. A may also refer to: Science and technology Quantities and units * ''a'', a measure for the attraction between particles in the Van der Waals equation * ''A'' value, a measure of ...
studied the patterns of phyllotaxis in his 1904 book. In 1917, D'Arcy Wentworth Thompson published '' On Growth and Form''; his description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants showed that simple equations could describe the spiral growth patterns of animal horns and mollusc shells. 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 c ...
(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. He predicted
oscillating Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
chemical reactions, in particular the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms can, Turing suggested, generate patterns (dubbed "
Turing pattern The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomousl ...
s") of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis. In 1968, the Hungarian theoretical biologist Aristid Lindenmayer (1925–1989) developed the L-system, 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 Grzegorz Rozenberg (born 14 March 1942, Warsaw) is a Polish and Dutch computer scientist. His primary research areas are natural computing, formal languages, formal language and automata theory, graph rewriting, graph transformations, and pe ...
; Salomaa, Arto. ''The Mathematical Theory of L Systems''. Academic Press, 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 s ...
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 mat ...
, Georg Cantor,
Helge von Koch Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described. He was born to Swedish nobility. ...
, Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, '' How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension'', crystallising mathematical thought into the concept of the fractal. File:Cycas circinalis male cone in Olomouc.jpg, Fibonacci number patterns occur widely in plants such as this queen sago, '' Cycas circinalis''. File:National Aquatics Center Construction (cropped).jpg, Beijing's National Aquatics Center for the 2008 Olympic games has a Weaire–Phelan structure. File:Drcy.svg, D'Arcy Thompson pioneered the study of growth and form in his 1917 book.


Causes

Living things like orchids, 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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-systems 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 rel ...
apply the abstractions of mathematics to the real world, often as if it were
perfect Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...
. 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, macr ...
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 laws; 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 ban ...
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 hereditar ...
,
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, and different kinds of signalling, including
mimicry In evolutionary biology, mimicry is an evolved resemblance between an organism and another object, often an organism of another species. Mimicry may evolve between different species, or between individuals of the same species. Often, mimicry f ...
and
cleaning symbiosis Cleaning symbiosis is a mutually beneficial association between individuals of two species, where one (the cleaner) removes and eats parasites and other materials from the surface of the other (the client). Cleaning symbiosis is well-known amon ...
. In plants, the shapes, colours, and patterns of insect-pollinated flowers like the lily 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as
sea anemone Sea anemones are a group of predatory marine invertebrates of the order Actiniaria. Because of their colourful appearance, they are named after the '' Anemone'', a terrestrial flowering plant. Sea anemones are classified in the phylum Cnidaria, ...
s. Fivefold symmetry is found in the echinoderms, the group that includes
starfish Starfish or sea stars are star-shaped echinoderms belonging to the class Asteroidea (). Common usage frequently finds these names being also applied to ophiuroids, which are correctly referred to as brittle stars or basket stars. Starfish ...
,
sea urchin Sea urchins () are spiny, globular echinoderms in the class Echinoidea. About 950 species of sea urchin live on the seabed of every ocean and inhabit every depth zone from the intertidal seashore down to . The spherical, hard shells (tests) of ...
s, and
sea lilies Crinoids are marine animals that make up the class Crinoidea. Crinoids that are attached to the sea bottom by a stalk in their adult form are commonly called sea lilies, while the unstalked forms are called feather stars or comatulids, which are ...
. Among non-living things, snowflakes 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, macr ...
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). 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 A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
shape and rings of a planet like
Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
. 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. File:Starfish 02 (paulshaffner) cropped.jpg, Echinoderms like this
starfish Starfish or sea stars are star-shaped echinoderms belonging to the class Asteroidea (). Common usage frequently finds these names being also applied to ophiuroids, which are correctly referred to as brittle stars or basket stars. Starfish ...
have fivefold symmetry. File:Medlar 5-symmetry.jpg, Fivefold symmetry can be seen in many flowers and some fruits like this medlar. File:Schnee2.jpg, Snowflakes have sixfold symmetry. File:Aragonite-Fluorite-cflu02c.jpg, Fluorite showing cubic crystal habit. File:Water splashes 001.jpg, Water splash approximates radial symmetry. 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 ...
showing rhombic dodecahedral crystal habit. File:Mikrofoto.de-volvox-8.jpg, '' Volvox'' has spherical symmetry. File:Two Oceans Aquarium03.jpg,
Sea anemone Sea anemones are a group of predatory marine invertebrates of the order Actiniaria. Because of their colourful appearance, they are named after the '' Anemone'', a terrestrial flowering plant. Sea anemones are classified in the phylum Cnidaria, ...
s have rotational symmetry.


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 tra ...
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 re ...
. 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 [below 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 Pythagorean triplet, right-angled triangle. One explanation is that this allows trees to better withstand high winds. Simulations of biomechanical models agree with the rule. Fractals are infinitely self-similarity, self-similar, iterated mathematical constructs having fractal dimension. Infinite iteration is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of ferns and umbellifers (Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including bryozoa, corals, hydrozoa like the air fern, ''Sertularia argentea'', and in non-living things, notably electrical discharges. L-system, Lindenmayer system fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points (Internode (botany), internode length), and number of branches per branch point. Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river, river networks, geologic fault lines, mountains, coastlines, animal coloration, snowflake, 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, macr ...
s, blood vessel branching, Purkinje cell, Purkinje cells, actin cytoskeleton, actin cytoskeletons, and wind wave, ocean waves. File:Dragon trees.jpg, The growth patterns of certain trees resemble these L-system, Lindenmayer system fractals. File:Baobab Tree at Vasai Fort.jpg, Branching pattern of a baobab tree File:Anthriscus sylvestris (Köhler's Medizinal-Pflanzen).jpg, Leaf of cow parsley, ''Anthriscus sylvestris'', is 2- or 3-pinnate, not infinite File:Romanesco broccoli (Brassica oleracea).jpg, Fractal spirals: Romanesco broccoli showing self-similar form File:Angelica flowerhead showing pattern.JPG, Angelica flowerhead, a sphere made of spheres (self-similar) File:Square1.jpg, Trees: Lichtenberg figure: high voltage dielectric breakdown in an Poly(methyl methacrylate), acrylic polymer block File:Dendritic Copper Crystals - 20x magnification.jpg, Trees: Dendrite (crystal), dendritic copper crystals (in microscope)


Spirals

Spirals are common in plants and in some animals, notably molluscs. For example, in the nautilus, a cephalopod mollusc, each Camera (cephalopod), chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. 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, the arrangement of leaves on a stem, and in the arrangement (parastichy) of other parts as in compositae, composite flower, flower heads and seed, seed heads like the sunflower or fruit structures like the pineapple and salak, snake fruit, as well as in the pattern of scales in pine cones, where multiple spirals run both clockwise and anticlockwise. These arrangements have explanations at different levels – mathematics, physics, chemistry, biology – each individually correct, but all necessary together. Phyllotaxis spirals can be generated from Fibonacci number, Fibonacci ratios: the Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13... (each subsequent number being the sum of the two preceding ones). For example, when leaves alternate up a stem, one rotation of the spiral touches two leaves, so the pattern or ratio is 1/2. In hazel the ratio is 1/3; in apricot it is 2/5; in pear it is 3/8; in almond it is 5/13. In disc phyllotaxis as in the sunflower and Bellis perennis, daisy, the florets are arranged along Fermat's spiral, but this is disguised because successive florets are spaced far apart, by the golden angle, 137.508° (dividing the circle in the
golden ratio 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 processes in dynamic systems. 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 proteins that manipulate the concentration of the plant hormone auxin, which activates meristem growth, alongside other mechanisms to control the relative angle of buds around the stem. 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. File:Fibonacci spiral 34.svg, Fibonacci number, Fibonacci spiral File:Ovis canadensis 2 (cropped).jpg, Bighorn sheep, ''Ovis canadensis'' File:Aloe polyphylla spiral.jpg, Spirals: phyllotaxis of spiral aloe, ''Aloe polyphylla'' File:NautilusCutawayLogarithmicSpiral.jpg, ''Nautilus'' shell's logarithmic growth spiral File:Pflanze-Sonnenblume1-Asio (cropped).JPG, Fermat's spiral: seed head of sunflower, ''Helianthus annuus'' File:Red Cabbage cross section showing spirals.jpg, Multiple Fibonacci spirals: red cabbage in cross section File:Trochoidea liebetruti (Albers, 1852) (4308584755).jpg, Spiralling shell of ''Trochoidea liebetruti'' File:Fibonacci spin (cropped).jpg, Water droplets fly off a wet, spinning ball in equiangular spirals


Chaos, flow, meanders

In mathematics, a dynamical system 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 set, dense periodic orbits. Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the ''strange attractors'' in chaotic systems have a fractal dimension. Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30. Vortex streets are zigzagging patterns of whirling Vortex, vortices created by the unsteady flow separation, separation of flow of a fluid, most often air or water, over obstructing objects. Smooth (Laminar flow, laminar) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the viscosity of the fluid. Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend. The outside of the loop is left clean and unprotected, so erosion accelerates, further increasing the meandering in a powerful positive feedback loop. File:Textile cone (cropped).JPG, Chaos: shell of gastropod mollusc the cloth of gold cone, ''Conus textile'', resembles Rule 30 cellular automaton File:Vortex-street-1.jpg, Flow: vortex street of clouds at Juan Fernandez Islands File:Rio Negro meanders.JPG, Meanders: dramatic meander 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 surface with minimal area ( minimal surface) — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one. A foam is a mass of bubbles; foams of different materials occur in nature. Foams composed of soap films obey Plateau's laws, which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the tetrahedron, tetrahedral angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant mean curvature, average curvature at every point. For example, a film may remain nearly flat on average by being curved up in one direction (say, left to right) while being curved downwards in another direction (say, front to back). 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 Sea urchins () are spiny, globular echinoderms in the class Echinoidea. About 950 species of sea urchin live on the seabed of every ocean and inhabit every depth zone from the intertidal seashore down to . The spherical, hard shells (tests) of ...
, ''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 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, macr ...
, 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 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 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'', 1202. ** ———— translated by Sigler, Laurence E. ''Fibonacci's Liber Abaci''. Springer, 2002. * Ernst Haeckel, Haeckel, Ernst. ''Kunstformen der Natur'' (Art Forms in Nature), 1899–1904. * D'Arcy Wentworth Thompson, Thompson, D'Arcy Wentworth. '' On Growth and Form''. Cambridge, 1917. General books * Adam, John A
''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