Patterns in nature

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. Natural patterns include symmetries, trees, spirals, waves, foams, tessellations, cracks and stripes. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.

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

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

History

Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. Pythagoras (c. 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 (c. 494–c. 434 BC) to an extent anticipated Darwin's evolutionary explanation for the structures of organisms. Plato (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 (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 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 of the quincunx 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 series. Mathematical observations of phyllotaxis followed with Karl Friedrich Schimper and his friend Alexander Braun's 1830 and 1830 work, respectively; Auguste Bravais and his brother Louis connected phyllotaxis ratios to the Fibonacci sequence in 1837, also noting its appearance in pinecones and pineapples. 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 crystals.

In the 19th century, the Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams. 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 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 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 (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 chemical reactions, in particular the Belousov–Zhabotinsky reaction. 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 (1925–1989) developed the L-system, a formal grammar 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, New York, 1980. L-systems have an alphabet 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, Georg Cantor, Helge von Koch, Wacław Sierpiński and others, Benoît Mandelbrot wrote a famous paper, '' How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension'', crystallising mathematical thought into the concept of the fractal.

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 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 apply the abstractions of mathematics to the real world, often as if it were perfect. For example, a crystal is perfect when it has no structural defects such as dislocations and is fully symmetric. Exact mathematical perfection can only approximate real objects. Visible patterns in nature are governed by physical laws; for example, meanders can be explained using fluid dynamics.

In biology, natural selection can cause the development of patterns in living things for several reasons, including camouflage,Darwin, Charles. ''On the Origin of Species''. 1859, chapter 4. sexual selection, and different kinds of signalling, including mimicry and cleaning symbiosis. 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 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 anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies.

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. Crystals 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 shape and rings of a planet 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.
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.
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 showing rhombic dodecahedral crystal habit.
File:Mikrofoto.de-volvox-8.jpg, '' Volvox'' has spherical symmetry.
File:Two Oceans Aquarium03.jpg, Sea anemones have rotational symmetry.

Trees, fractals

The branching pattern of trees was described in the Italian Renaissance by Leonardo da Vinci. 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. 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. 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. 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 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 lines, mountains, coastlines, animal coloration, snow flakes, crystals, blood vessel branching, Purkinje cells, actin cytoskeletons, and ocean waves.

File:Dragon trees.jpg, The growth patterns of certain trees resemble these 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 acrylic polymer block
File:Dendritic Copper Crystals - 20x magnification.jpg, Trees: dendritic copper crystals (in microscope)

Spirals

Spirals