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Patterns in nature

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Patterns in nature are regularities of visual form found in the natural world. These patterns, such as spirals, trees, arrays, and cracks recur in many different contexts. Each such pattern can be modelled mathematically.[1] Patterns in living things can be explained at different levels by mathematics, physics, chemistry, and natural selection. Each of these can be considered 'causes' of pattern.

Natural patterns being formed: sand blowing into dunes in Namibia. The crescent shaped dunes and the ripples on their surfaces repeat wherever there are suitable conditions

History

The Bible mentions the Lily's beautiful patterns[2]

Plato

The Greek philosopher Plato (ca 427 – ca 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") in his philosophy, Platonic realism. Platonic forms are perfect abstract objects or (in a sense) patterns in nature; physical objects are never more than imperfect copies of the ideal forms. Thus, a flower may be roughly circular, but it is never a perfect, mathematical circle.[3]

Fibonacci

In 1202, Leonardo Fibonacci (ca 1170 – ca 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci.[4]

D'Arcy Thompson

D'Arcy Thompson pioneered the study of growth and form

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, are classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns and mollusc shells.[5]

Turing

In 1952, Alan Turing (1912–1954), better known for his work on computing and codebreaking, wrote The Chemical Basis of Morphogenesis.[6] He predicted oscillating chemical reactions, in particular the Belousov–Zhabotinsky reaction. Such activator-inhibitor mechanisms can generate patterns of stripes and spots, and contribute to the spiral patterns seen in plant phyllotaxis.[7]

Lindenmayer, Mandelbrot

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.[8] 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, crystalling mathematical thought into the concept of the fractal.[9]

Causes

The growth patterns of trees resemble these Lindenmayer system fractals

Peter Forbes wrote in The Guardian that "Orchids, hummingbirds, the peacock's tail have a beauty of form, pattern and colour that artists struggle to match – and these are abstract designs." [10]

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.[11]

Mathematics

Mathematics seeks to discover and explain abstract patterns or regularities of all kinds.[12] [13] 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.[8]

Physics

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. Real objects can only approximate to such exact mathematical perfection.[14] Visible patterns in nature are governed by physical laws such as of fluid dynamics.

Natural selection

The laws of biology, particularly natural selection, can cause the development of patterns in living things for several reasons, including camouflage,[15] sexual selection,[15] and different kinds of signalling, such as mimicry[16] and cleaning symbiosis.[17] In plants, the shapes, colours, and patterns of many flowers like the lily have evolved to optimise insect pollination (other plants may be pollinated by wind, birds, or bats). Pollinating insects such as the European honey bee are attracted to such flowers by a radial pattern of colours and stripes (some visible only in ultraviolet light) that serve as nectar guides that can be seen at a distance; by scent; and by rewards such as sugar-rich nectar and edible pollen.[18]

Types of pattern

Trees, fractals

Mathematically, fractals are infinitely self-similar, iterated, constructs having fractal dimensions.[9][19][20] In nature, infinite iteration is not possible so all 'fractal' patterns are only approximate. For example, the leaves of plants such as ferns and umbellifers are only self-similar (pinnate) to 2, 3 or 4 levels.

Fern-like growth patterns occur not only in plants but in animals such as bryozoa, corals, hydrozoa like the air fern Sertularia argentea, and in non-living things such as electrical discharges. In fact, fractal-like patterns occur very widely in nature, in phenomena including clouds, river networks, geologic fault lines, mountains, coastlines,[9] animal coloration, snow flakes,[21] crystals[22] blood vessel branching,[23] ocean waves[24] and many others.

Spirals

Spirals are common in plants and in some animals, notably molluscs. For example, each chamber of the shell of a nautilus, a cephalopod mollusc, is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral.[25] 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 of other parts such as in composite flower heads and seed heads like the sunflower or fruit structures like the pineapple[26]: 337  and snakefruit, as well as in the pattern of scales in pinecones, 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.[27]

Phyllotaxis spirals can be generated mathematically from 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.[28]

From the point of view of physics, spirals are lowest-energy configurations.[29] They emerge spontaneously through self-organizing processes in dynamic systems.[30]

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.[31]

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.[26]

Chaos, flow, meanders

In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect"[32]), which requires the mathematical properties of topological mixing and dense periodic orbits.[33]

Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature. In fact there is a relationship between chaos and fractals; the strange attractors in chaotic systems have a fractal dimension.[34] Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's rule 30.[35]

Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid such as air or water over obstructing objects.[36] Smooth (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. They 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 such as sand 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.[37]

Waves, dunes

Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by.[38] 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.[39] 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.[40]

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 on to 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.[41] Apart from this nonlinearity, barchans behave rather like solitary waves.[42]

  • Waves: Breaking wave in a ship's wake
    Waves: Breaking wave in a ship's wake
  • Waves: Ripples in sand, with dislocations near centre of image
    Waves: Ripples in sand, with dislocations near centre of image
  • Dunes: sand dunes in Taklamakan desert, from space
    Dunes: sand dunes in Taklamakan desert, from space
  • Dunes: Barchan crescent sand dune
    Dunes: Barchan crescent sand dune

Arrays, crystals, tilings

The structures of minerals provide many examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, here are rather few possible types of arrangement of atoms in a crystal: see crystal structure, crystal system, and point group for details.

Tessellations are patterns formed by repeating Tiles all over a flat surface. There are 17 wallpaper groups of such tilings.[43] 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 honey bee's wax cells in honeycomb are well-known examples. When animals such as 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. Some flowers such as the Snakeshead Fritillary, Fritillaria meleagris, have a tessellated chequerboard pattern on their petals.

Cracks

Cracks form in materials to relieve stress. When an 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.[44]

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.[45]

  • Old pottery surface, white glaze with mainly 90° cracks
    Old pottery surface, white glaze with mainly 90° cracks
  • Cracks: Drying inelastic mud in the Rann of Kutch with mainly 90° cracks
    Cracks: Drying inelastic mud in the Rann of Kutch with mainly 90° cracks
  • Veined Gabbro with 90° cracks, near Sgurr na Stri, Skye
    Veined Gabbro with 90° cracks, near Sgurr na Stri, Skye
  • Cracks: Drying elastic mud in Sicily with mainly 120° cracks
    Cracks: Drying elastic mud in Sicily with mainly 120° cracks
  • Cooled basalt at Giant's Causeway. Vertical mainly 120° cracks giving hexagonal columns
    Cooled basalt at Giant's Causeway. Vertical mainly 120° cracks giving hexagonal columns
  • Cracks: tough tree bark splits between fibres as tree grows
    Cracks: tough tree bark splits between fibres as tree grows
  • Cracks: Palm trunk with branching vertical cracks (and horizontal leaf scars)
    Cracks: Palm trunk with branching vertical cracks (and horizontal leaf scars)

See also

Bibliography

Pioneering authors

Latin Wikisource has original text related to this article:
  • Fibonacci, Leonardo. Liber Abaci, 1202.
    • ----- translated by Sigler, Laurence E. Fibonacci's Liber Abaci. Springer, 2002.
  • 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, 2006.
  • Ball, Philip. Nature's Patterns: a tapestry in three parts. 1:Shapes. 2:Flow. 3:Branches. Oxford, 2009.
  • Murphy, Pat. By Nature's Design.
  • Rothenburg, David. Survival of the Beautiful: Art, Science and Evolution. Bloomsbury, 2011.
  • Stevens, Peter S. Patterns in Nature. Little, Brown & Co, 1974.
  • Stewart, Ian. What Shape is a Snowflake? Weidenfeld & Nicolson, 2001.

Patterns from nature (as art)

  • Edmaier, Bernard. Patterns of the Earth. Phaidon, 2007.
  • Porter, Eliot, and Gleick, James. Nature's Chaos. Viking, 1990.
  • 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.

References

  1. ^ Peter Stevens. Patterns in Nature, 1974. Page 3.
  2. ^ And why take ye thought for raiment? Consider the lilies of the field, how they grow; they toil not, neither do they spin:
    And yet I say unto you, That even Solomon in all his glory was not arrayed like one of these    . King James Bible, 1769. Matthew 6:28–29.
  3. ^ Balaguer, Mark (12 May 2004, revised 7 April 2009). "Stanford Encyclopedia of Philosophy". Platonism in Metaphysics. Stanford University. Retrieved 4 May 2012. {{cite web}}: Check date values in: |date= (help)
  4. ^ Parmanand Singh. Acharya Hemachandra and the (so called) Fibonacci Numbers. Math. Ed. Siwan , 20(1):28–30, 1986. ISSN 0047-6269
  5. ^ About D'Arcy Retrieved 29 April 2012.
  6. ^ Turing, A. M. (1952). "The Chemical Basis of Morphogenesis". Philosophical Transactions of the Royal Society of London, series B. 237 (641): 37–72. doi:10.1098/rstb.1952.0012. {{cite journal}}: Invalid |ref=harv (help)
  7. ^ Philip Ball. Shapes. Pages 163, 247–250.
  8. ^ a b Grzegorz Rozenberg and Arto Salomaa. The mathematical theory of L systems. Academic Press, New York, 1980. ISBN 0-12-597140-0
  9. ^ a b c Mandelbrot,Benoît B. (1983). The fractal geometry of nature. Macmillan. Cite error: The named reference "Mandelbrot" was defined multiple times with different content (see the help page).
  10. ^ Forbes, Peter. All that useless beauty. The Guardian. Review: Non-fiction. 11 February 2012.
  11. ^ Peter Stevens, 1994. Page 222.
  12. ^ Steen, L.A.. The Science of Patterns. Science, 240: 611–616, 1998. Summary at ascd.org
  13. ^ Devlin, Keith. Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996
  14. ^ Tatarkiewicz, Perfection in the Sciences. II. Perfection in Physics and Chemistry, Dialectics and Humanism, vol. VII, no. 2 (spring 1980), p. 139.
  15. ^ a b Darwin, Charles. On the Origin of Species. 1859, chapter 4.
  16. ^ Wickler, W. (1968). Mimicry in plants and animals. New York: McGraw-Hill.
  17. ^ Poulin, R., Grutter, A. S., (1996) "Cleaning symbiosis: proximate and adaptive explanations". Bioscience 46(7): 512-517. http://www.lexagrutter.com/Publications/paper%207.pdf
  18. ^ Koning, Ross (1994). "Plant Physiology Information Website". Pollination Adaptations. Ross Koning. Retrieved May 02, 2012. {{cite web}}: Check date values in: |accessdate= (help)
  19. ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.
  20. ^ Briggs, John (1992). Fractals:The Patterns of Chaos. London : Thames and Hudson, 1992. p. 148. {{cite book}}: Cite has empty unknown parameter: |coauthors= (help)
  21. ^ Yves Meyer and Sylvie Roques (1993). Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications," Toulouse, France – June 1992. Atlantica Séguier Frontières. p. 25.
  22. ^ Alessandra Carbone, Mikhael Gromov, Przemyslaw Prusinkiewicz (2000). Pattern formation in biology, vision and dynamics. World Scientific. p. 78. ISBN 9789810237929.{{cite book}}: CS1 maint: multiple names: authors list (link)
  23. ^ Horst K. Hahn; Manfred Georg; and Heinz-Otto Peitgen (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. (ed.). Fractals in biology and medicine. Springer. pp. 55–66.{{cite book}}: CS1 maint: multiple names: authors list (link)>
  24. ^ Paul S. Addison (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44–46.
  25. ^ Eli Maor. e: The Story of a Number. Princeton University Press, 2009. Page 135.
  26. ^ a b Kappraff, Jay (2004). "Growth in Plants: A Study in Number" (PDF). Forma. 19: 335–354.
  27. ^ Philip Ball. Shapes. Page 13.
  28. ^ Coxeter, H. S. M. (1961). Introduction to geometry. Wiley. p. 169.
  29. ^ Levitov LS (15 March 1991). "Energetic Approach to Phyllotaxis" (PDF). Europhys. Lett. 14 (6): 533–9. doi:10.1209/0295-5075/14/6/006.
  30. ^ Douady S, Couder Y (1992). "Phyllotaxis as a physical self-organized growth process". Phys. Rev. Lett. 68 (13): 2098–2101. Bibcode:1992PhRvL..68.2098D. doi:10.1103/PhysRevLett.68.2098. PMID 10045303. {{cite journal}}: Unknown parameter |month= ignored (help)
  31. ^ Philip Ball. Shapes. Pages 163, 249–250.
  32. ^ Lorenz, Edward N. (1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. ISSN 1520-0469. Retrieved 3 June 2010. {{cite journal}}: Unknown parameter |month= ignored (help)
  33. ^ Elaydi, Saber N. (1999). Discrete Chaos. Chapman & Hall/CRC. p. 117.
  34. ^ David Ruelle. Chance and Chaos. Princeton University Press, 1991.
  35. ^ Wolfram, Stephen, A New Kind of Science. Wolfram Media, 2002.
  36. ^ Theodore von Kármán, Aerodynamics. McGraw-Hill (1963): ISBN 978-0070676022. Dover (1994): ISBN 978-0486434858.
  37. ^ Lewalle, Jacques (2006). "Flow Separation and Secondary Flow: Section 9.1". Lecture Notes in Incompressible Fluid Dynamics: Phenomenology, Concepts and Analytical Tools (PDF). Syracuse, NY: Syracuse University..
  38. ^ French, A.P. Vibrations and Waves. Nelson Thornes, 1971.
  39. ^ Tolman, H.L. (2008), "Practical wind wave modeling", in Mahmood, M.F. (ed.), CBMS Conference Proceedings on Water Waves: Theory and Experiment (PDF), Howard University, USA, 13–18 May 2008: World Scientific Publ.{{citation}}: CS1 maint: location (link)
  40. ^ "Types of Dunes". USGS. 29 October 1997. Retrieved May 02, 2012. {{cite web}}: Check date values in: |accessdate= (help)
  41. ^ Strahler, A. & Archibold, O.W. Physical Geography: Science and Systems of the Human Environment. John Wiley, 4th edition 2008. Page 442.
  42. ^ Schwämmle, V., and H.J. Herrmann (2003). "Solitary wave behaviour of sand dunes". Nature. 426 (Dec. 11): 619–620 Abstract. doi:10.1038/426619a. PMID 14668849.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  43. ^ Armstrong, M.A. (1988). Groups and Symmetry. New York: Springer-Verlag.
  44. ^ Peter Stevens, 1974. Page 207.
  45. ^ Peter Stevens, 1974. Page 208.

External links

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