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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, tessellations, cracks and stripes.〔Stevens, Peter. ''Patterns in Nature'', 1974. Page 3.〕 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. ==History== 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 ( ''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. Pythagoras explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.〔The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things. Aristotle, ''Metaphysics 1–5 '', cc. 350 BC〕 Empedocles to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.〔Aristotle reports Empedocles arguing that, "()herever, then, everything turned out as it would have if it were happening for a purpose, there the creatures survived, being accidentally compounded in a suitable way; but where this did not happen, the creatures perished." ''The Physics'', B8, 198b29 in Kirk, et al., 304).〕 In 1202, Leonardo Fibonacci (c 1170 – c 1250) introduced the Fibonacci number sequence to the western world with his book ''Liber Abaci''.〔Singh, Parmanand. ''Acharya Hemachandra and the (so called) Fibonacci Numbers''. Math. Ed. Siwan, 20(1):28–30, 1986. 〕 Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population. 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.〔(About D'Arcy ). D' Arcy 150. University of Dundee and the University of St Andrews. Retrieved 16 October 2012.〕 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.〔Stewart, Ian. 2001. Pages 108–109.〕 The German psychologist Adolf Zeising (1810–1876) claimed that the golden ratio was expressed in the arrangement of plant parts, in the skeletons of animals and the branching patterns of their veins and nerves, as well as in the geometry of crystals. Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their symmetry to support his faux-Darwinian theories of evolution.〔Ball, Philip. ''Shapes''. 2009. Page 41.〕 The American photographer Wilson Bentley (1865–1931) took the first micrograph of a snowflake in 1885. 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. 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.〔Ball, Philip. ''Shapes''. 2009. Pages 163, 247–250.〕 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.〔Rozenberg, Grzegorz; Salomaa, Arto. ''The mathematical theory of L systems''. Academic Press, New York, 1980. ISBN 0-12-597140-0〕 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.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Patterns in nature」の詳細全文を読む スポンサード リンク
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