Patterns in nature

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Namib Desert. The crescent shaped dunes and the ripples
on their surfaces repeat wherever there are suitable conditions.
Patterns of the veiled chameleon, Chamaeleo calyptratus, provide camouflage and signal mood as well as breeding condition.

Patterns in nature are visible regularities of form found in the

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

fractals
could create plant growth patterns.

computer models
to simulate a wide range of patterns.

History

Early Greek philosophers attempted to explain order in

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 circle.[2] 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.[3] 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.[4][3]

In 1202,

pinecones and pineapples.[3] 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.[8][9][10]

In the 19th century, the Belgian physicist

Beijing National Aquatics Center adapted the structure for their outer wall in the 2008 Summer Olympics.[12] Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their symmetry to support his faux-Darwinian theories of evolution.[13] The American photographer Wilson Bentley took the first micrograph of a snowflake in 1885.[14]

In the 20th century,

animal horns and mollusc shells.[16] 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.[17] 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.[18]
In 1968, the Hungarian theoretical biologist
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, crystallising mathematical thought into the concept of the fractal.[20]

  • Fibonacci number patterns occur widely in plants such as this queen sago, Cycas circinalis.
    Fibonacci number patterns occur widely in plants such as this queen sago, Cycas circinalis
    .
  • Beijing's National Aquatics Center for the 2008 Olympic games has a Weaire–Phelan structure.
    Beijing's National Aquatics Center for the 2008 Olympic games has a Weaire–Phelan structure.
  • D'Arcy Thompson pioneered the study of growth and form in his 1917 book.
    D'Arcy Thompson pioneered the study of growth and form in his 1917 book.

Causes

veins
, which are avoided by the young aphids

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.[21] 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.[22]

Mathematics seeks to discover and explain abstract patterns or regularities of all kinds.[23][24] 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.[19]

The laws of

physical laws; for example, meanders can be explained using fluid dynamics
.

In

nectar guides that can be seen at a distance.[29]

Types of pattern

Symmetry

Further information:
Crystal symmetry

sea lilies.[31]

Among non-living things,

spheroidal shape and rings of a planet like Saturn.[35]

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).[36] 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.[37] In the case of ice eggs, the gentle churn of water, blown by a suitably stiff breeze makes concentric layers of ice form on a seed particle that then grows into a floating ball as it rolls through the freezing currents.[38]

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 [below them].[39]

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.

right-angled triangle. One explanation is that this allows trees to better withstand high winds.[40] Simulations of biomechanical models agree with the rule.[41]

internode length), and number of branches per branch point.[19]

Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds,

actin cytoskeletons,[49] and ocean waves.[50]

Spirals

Further information: Phyllotaxis

molluscs. For example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral.[51] Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity.[52]

Plant spirals can be seen in

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.[56]

In disc phyllotaxis as in the

sunflower and daisy, the florets are arranged along Fermat's spiral, but this is disguised because successive florets are spaced far apart, by the golden angle, 137.508° (dividing the circle in the golden ratio); when the flowerhead is mature so all the elements are the same size, this spacing creates a Fibonacci number of more obvious spirals.[57]

From the point of view of physics, spirals are lowest-energy configurations

dynamic systems.[59] 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.[60] 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.[54]

Chaos, flow, meanders

In mathematics, a

periodic orbits.[62]

Alongside fractals,

cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30.[64]

Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects.[65] 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.

positive feedback loop.[66]

Waves, dunes

Taklamakan desert. Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or seif ('sword') shapes.[69]

slip face point downwind. Sand blows over the upwind face, which stands at about 15 degrees from the horizontal, and falls onto 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.[70] Apart from this nonlinearity, barchans behave rather like solitary waves.[71]

  • Waves: breaking wave in a ship's wake
    Waves: breaking wave in a ship's wake
  • 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
  • Wind ripples with dislocations in Sistan, Afghanistan
    Wind ripples with dislocations in Sistan, Afghanistan

Bubbles, foam

A soap bubble forms a sphere, a surface with minimal area (minimal surface) — the smallest possible surface area for the volume enclosed. Two bubbles together form a more complex shape: the outer surfaces of both bubbles are spherical; these surfaces are joined by a third spherical surface as the smaller bubble bulges slightly into the larger one.[11]

A foam is a mass of bubbles; foams of different materials occur in nature. Foams composed of soap films obey Plateau's laws, which require three soap films to meet at each edge at 120° and four soap edges to meet at each vertex at the tetrahedral angle of about 109.5°. Plateau's laws further require films to be smooth and continuous, and to have a constant average curvature at every point. For example, a film may remain nearly flat on average by being curved up in one direction (say, left to right) while being curved downwards in another direction (say, front to back).[72][73] Structures with minimal surfaces can be used as tents.

At the scale of living

convex polyhedron, the number of faces plus the number of vertices (corners) equals the number of edges plus two. A result of this formula is that any closed polyhedron of hexagons has to include exactly 12 pentagons, like a soccer ball, Buckminster Fuller geodesic dome, or fullerene molecule. This can be visualised by noting that a mesh of hexagons is flat like a sheet of chicken wire, but each pentagon that is added forces the mesh to bend (there are fewer corners, so the mesh is pulled in).[76]

Tessellations

Main article: Tessellation

Bravais lattices for the 7 lattice systems in three-dimensional space.[78]

Cracks

Cracks are linear openings that 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.[79] 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.[80]

  • Old pottery surface, white glaze with mainly 90° cracks
    Old pottery surface, white glaze with mainly 90° cracks
  • Drying inelastic mud in the Rann of Kutch with mainly 90° cracks
    Drying inelastic mud in the Rann of Kutch with mainly 90° cracks
  • Veined gabbro with 90° cracks, near Sgurr na Stri, Skye
    Veined
    Skye
  • Drying elastic mud in Sicily with mainly 120° 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
  • Palm trunk with branching vertical cracks (and horizontal leaf scars)
    Palm trunk with branching vertical cracks (and horizontal leaf scars)

Spots, stripes

Leopards and ladybirds are spotted; angelfish and zebras are striped.

predatory birds that hunt by sight, if it has bold warning colours, and is also distastefully bitter or poisonous, or mimics other distasteful insects. A young bird may see a warning patterned insect like a ladybird and try to eat it, but it will only do this once; very soon it will spit out the bitter insect; the other ladybirds in the area will remain undisturbed. The young leopards and ladybirds, inheriting genes that somehow create spottedness, survive. But while these evolutionary and functional arguments explain why these animals need their patterns, they do not explain how the patterns are formed.[81]

Pattern formation

Main article: Pattern formation

Alan Turing,[17] and later the mathematical biologist James Murray,[82] described a mechanism that spontaneously creates spotted or striped patterns: a reaction–diffusion system.[83] The cells of a young organism have genes that can be switched on by a chemical signal, a morphogen, resulting in the growth of a certain type of structure, say a darkly pigmented patch of skin. If the morphogen is present everywhere, the result is an even pigmentation, as in a black leopard. But if it is unevenly distributed, spots or stripes can result. Turing suggested that there could be feedback control of the production of the morphogen itself. This could cause continuous fluctuations in the amount of morphogen as it diffused around the body. A second mechanism is needed to create standing wave patterns (to result in spots or stripes): an inhibitor chemical that switches off production of the morphogen, and that itself diffuses through the body more quickly than the morphogen, resulting in an activator-inhibitor scheme. The Belousov–Zhabotinsky reaction is a non-biological example of this kind of scheme, a chemical oscillator.[83]

Later research has managed to create convincing models of patterns as diverse as zebra stripes, giraffe blotches, jaguar spots (medium-dark patches surrounded by dark broken rings) and ladybird shell patterns (different geometrical layouts of spots and stripes, see illustrations).

Numida meleagris in which the individual feathers feature transitions from bars at the base to an array of dots at the far (distal) end. These require an oscillation created by two inhibiting signals, with interactions in both space and time.[86]

Patterns can form for other reasons in the

fairy circles of Namibia appear to be created by the interaction of competing groups of sand termites, along with competition for water among the desert plants.[90]

In permafrost soils with an active upper layer subject to annual freeze and thaw,

Thermal contraction causes shrinkage cracks to form; in a thaw, water fills the cracks, expanding to form ice when next frozen, and widening the cracks into wedges. These cracks may join up to form polygons and other shapes.[91]

The

cortex. Similar patterns of gyri (peaks) and sulci (troughs) have been demonstrated in models of the brain starting from smooth, layered gels, with the patterns caused by compressive mechanical forces resulting from the expansion of the outer layer (representing the cortex) after the addition of a solvent. Numerical models in computer simulations support natural and experimental observations that the surface folding patterns increase in larger brains.[92][93]

See also

References

Footnotes

  1. ^ 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 , c. 350 BC
  2. ^ Aristotle reports Empedocles arguing that, "[w]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).

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Bibliography

Pioneering authors

Latin Wikisource has original text related to this article:

General books

Patterns from nature (as art)

  • Edmaier, Bernard. Patterns of the Earth. Phaidon Press, 2007.
  • Macnab, Maggie. Design by Nature: Using Universal Forms and Principles in Design. New Riders, 2012.
  • 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.
  • Viking Penguin
    , 1990.

External links

Retrieved from "https://en.wikipedia.org/w/index.php?title=Patterns_in_nature&oldid=1211021644"