He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams. 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. 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. Mathematical observations of phyllotaxis followed with Karl Friedrich Schimper and his friend Alexander Braun's 18 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 1754, Charles Bonnet observed that the spiral phyllotaxis of plants were frequently expressed in both clockwise and counter-clockwise golden ratio series. The discourse's central chapter features examples and observations of the quincunx in botany. 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. Johannes Kepler (1571–1630) pointed out the presence of the Fibonacci sequence in nature, using it to explain the pentagonal form of some flowers. Fibonacci presented a thought experiment on the growth of an idealized rabbit population. In 1202, Leonardo Fibonacci introduced the Fibonacci sequence to the western world with his book Liber Abaci. 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. 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. Thus, a flower may be roughly circular, but it is never a perfect circle. He considered these to consist of ideal forms ( εἶδος eidos: "form") of which physical objects are never more than imperfect copies. Plato (c. 427–c. 347 BC) argued for the existence of natural universals. Empedocles (c. 494–c. 434 BC) to an extent anticipated Darwin's evolutionary explanation for the structures of organisms. 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. Studies of pattern formation make use of computer models to simulate a wide range of patterns.Įarly Greek philosophers attempted to explain order in nature, anticipating modern concepts. Patterns in living things are explained by the biological processes of natural selection and sexual selection. Mathematics, physics and chemistry can explain patterns in nature at different levels and scales. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The modern understanding of visible patterns developed gradually over time. Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. These patterns recur in different contexts and can sometimes be modelled mathematically. Patterns in nature are visible regularities of form found in the natural world. Patterns of the veiled chameleon, Chamaeleo calyptratus, provide camouflage and signal mood as well as breeding condition. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions. Natural patterns form as wind blows sand in the dunes of the Namib Desert. Visible regularity of form found in the natural world
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