Tuesday, July 2, 2013

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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. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, arrays, cracks and stripes.[1] Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time.
In the 19th century, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. 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, British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Hungarian biologist Aristid Lindenmayer and 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. 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.
Contents  [hide] 
1 History
2 Causes
3 Types of pattern
3.1 Symmetry
3.2 Trees, fractals
3.3 Spirals
3.4 Chaos, flow, meanders
3.5 Waves, dunes
3.6 Bubbles, foam
3.7 Arrays, crystals, tilings
3.8 Cracks
3.9 Spots, stripes
3.10 Pattern formation
4 See also
5 Bibliography
5.1 Pioneering authors
5.2 General books
5.3 Patterns from nature (as art)
6 References
7 External links

Fibonacci patterns occur widely in plant structures including this cone of Queen sago, Cycas circinalis
Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. Plato (c 427 – c 347 BC) — looking only at his work on natural patterns — argued for the existence of universals. He considered these to consist of ideal forms (εἶδος eidos: "form") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect mathematical circle.[2] Pythagoras explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.[3] Empedocles to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.[4]
In 1202, Leonardo Fibonacci (c 1170 – c 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci.[5] Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population.[6] In 1917, D'Arcy Wentworth Thompson (1860–1948) published his book On Growth and Form. His description of phyllotaxis and the Fibonacci sequence, the mathematical relationships in the spiral growth patterns of plants, is classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns and mollusc shells.[7]
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.[8]
Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their symmetry to support his faux-Darwinian theories of evolution.[9]

D'Arcy Thompson pioneered the study of growth and form
In 1952, Alan Turing (1912–1954), better known for his work on computing and codebreaking, 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.[10] He predicted oscillating chemical reactions, in particular the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms can, Turing suggested, generate patterns of stripes and spots in animals, and contribute to the spiral patterns seen in plant phyllotaxis.[11]
In 1968, 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.[12] 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.[13]

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