Pattern formation

The science of pattern formation deals with the visible, (

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. The role of genes in pattern formation is an aspect of morphogenesis, the creation of diverse anatomies from similar genes, now being explored in the science of evolutionary developmental biology or evo-devo. The mechanisms involved are well seen in the anterior-posterior patterning of embryos from the model organism Drosophila melanogaster (a fruit fly), one of the first organisms to have its morphogenesis studied, and in the eyespots of butterflies, whose development is a variant of the standard (fruit fly) mechanism.

Patterns in nature

Examples of pattern formation can be found in biology, physics, and science,^[1] and can readily be simulated with computer graphics, as described in turn below.

Biology

Biological patterns such as

Vegetation patterns such as tiger bush^[16] and fir waves^[17] form for different reasons. Tiger bush consists of stripes of bushes on arid slopes in countries such as Niger where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation absorbs rainwater from the bare zone immediately above it.^[16] In contrast, fir waves occur in forests on mountain slopes after wind disturbance, during regeneration. When trees fall, the trees that they had sheltered become exposed and are in turn more likely to be damaged, so gaps tend to expand downwind. Meanwhile, on the windward side, young trees grow, protected by the wind shadow of the remaining tall trees.^[17] In flat terrains additional pattern morphologies appear besides stripes - hexagonal gap patterns and hexagonal spot patterns. Pattern formation in this case is driven by positive feedback loops between local vegetation growth and water transport towards the growth location.^[18][19]

Chemistry

This section needs expansion. You can help by adding to it. (March 2013)

Further information:

Turing instability.^[21] Pattern formation in chemical systems often involve oscillatory chemical kinetics or autocatalytic reactions^[22] such as Belousov–Zhabotinsky reaction or Briggs–Rauscher reaction. In industrial applications such as chemical reactors, pattern formation can lead to temperature hot spots which can reduce the yield or create hazardous safety problems such as a thermal runaway.^[23][20] The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying reaction-diffusion system.^[20][22]

Similarly as in chemical systems, patterns can develop in a weakly ionized plasma of a positive column of a glow discharge. In such cases creation and annihilation of charged particles due to collisions of atoms corresponds to reactions in chemical systems. Corresponding processes are essentially non-linear and lead in a discharge tube to formation of striations with regular or random character.[24]^[25]

• Belousov–Zhabotinsky reaction
• Liesegang rings
• Ionization waves

Physics

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When a planar body of fluid under the influence of gravity is heated from below, Rayleigh-Bénard convection can form organized cells in hexagons or other shapes. These patterns form on the

surface of the Sun and in the mantle of the Earth as well as during more pedestrian processes. The interaction between rotation, gravity, and convection can cause planetary atmospheres to form patterns, as is seen in Saturn's hexagon and the Great Red Spot and stripes of Jupiter. The same processes cause ordered cloud formations on Earth, such as stripes and rolls

.

In the 1980s Lugiato and Lefever developed a model of light propagation in an optical cavity that results in pattern formation by the exploitation of nonlinear effects.

Precipitating and solidifying materials can crystallize into intricate patterns, such as those seen in snowflakes and dendritic crystals.

Mathematics

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Sphere packings and coverings. Mathematics underlies the other pattern formation mechanisms listed.

Further information: Gradient pattern analysis

Computer graphics

reaction–diffusion

model, produced using sharpen and blur

Further information: Cellular automaton

Some types of

textures for more realistic shading of 3d objects.^[26][27]

A popular Photoshop plugin, KPT 6, included a filter called 'KPT reaction'. Reaction produced reaction–diffusion style patterns based on the supplied seed image.

A similar effect to the 'KPT reaction' can be achieved with convolution functions in digital image processing, with a little patience, by repeatedly sharpening and blurring an image in a graphics editor. If other filters are used, such as emboss or edge detection, different types of effects can be achieved.

Computers are often used to

reaction–diffusion or MClone

are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

References

1. ^ Ball, 2009.
2. ^ Ball, 2009. Shapes, pp. 231–252.
3. ^ Ball, 2009. Shapes, pp. 261–290.
4. S2CID 6930563
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5. S2CID 15998615
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6. ^ Hans Meinhard (2001-10-26). "Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation". Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany.
7. PMID 4390734
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8. ISBN 978-0-19-927536-6
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9. PMID 14756324
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10. ^ S. Kondo, T. Miura, "Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation", Science 24 Sep 2010: Vol. 329, Issue 5999, pp. 1616-1620 DOI: 10.1126/science.1179047
11. PMID 27145826
    .
12. ^ Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632
13. ^ Ball, 2009. Branches, pp. 52–59.
14. ^ Ball, 2009. Shapes, pp. 149–151.
15. PMID 33449631
    .
16. ^
    ISBN 978-1-4612-6559-7.{{cite book}}: CS1 maint: multiple names: authors list (link
    )
17. ^ ^{a b} D'Avanzo, C. (22 February 2004). "Fir Waves: Regeneration in New England Conifer Forests". TIEE. Retrieved 26 May 2012.
18. S2CID 209478350
    .
19. doi:10.1146/annurev-conmatphys-033117-053959
    .
20. ^
    ISSN 1385-8947
    .
21. ISBN 978-94-009-6239-2
22. ^
    S2CID 95837792
    .
23. ISSN 1520-6106
    .
24. doi:10.1063/1.1694626
    .
25. doi:10.1016/S0375-9601(01)00406-6
    .
26. ^ Greg Turk, Reaction–Diffusion
27. S2CID 207162368.{{cite book}}: CS1 maint: date and year (link
    )

Bibliography

• ISBN 978-0-19-960486-9
  .

External links

• SpiralZoom.com, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination (archived 14 May 2019)
• '15-line Matlab code', A simple 15-line Matlab program to simulate 2D pattern formation for reaction-diffusion model.