“斑图生成”的版本间的差异

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{{进一步|自然界中的模式}}
 
{{进一步|自然界中的模式}}
  
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Examples of pattern formation can be found in biology, chemistry, physics, and mathematics,<ref>Ball, 2009.</ref> and can readily be simulated with computer graphics, as described in turn below.
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Examples of pattern formation can be found in biology, chemistry, physics, and mathematics,<ref name=":2">Ball, 2009.</ref> and can readily be simulated with computer graphics, as described in turn below.
  
 
Examples of pattern formation can be found in biology, chemistry, physics, and mathematics, and can readily be simulated with computer graphics, as described in turn below.
 
Examples of pattern formation can be found in biology, chemistry, physics, and mathematics, and can readily be simulated with computer graphics, as described in turn below.
  
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在生物、化学、物理和数学中都有斑图生成的实例，并且我们可以用'''<font color="#ff8000"> 计算机图形学 Computer Graphics</font>''' 轻松地来模拟，下面依次进行介绍。
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在生物、化学、物理和数学<ref name=":2" /> 中都有斑图生成的实例，并且我们可以用'''<font color="#ff8000"> 计算机图形学 Computer Graphics</font>''' 轻松地来模拟，下面依次进行介绍。
  
 
===Biology 生物学===
 
===Biology 生物学===
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{{深入研究|进化发育生物学|形态发生学领域}}
 
{{深入研究|进化发育生物学|形态发生学领域}}
  
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Biological patterns such as [[animal markings]], the segmentation of animals, and [[phyllotaxis]] are formed in different ways.<ref>Ball, 2009. ''Shapes'', pp. 231–252.</ref>
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Biological patterns such as [[animal markings]], the segmentation of animals, and [[phyllotaxis]] are formed in different ways.<ref name=":3">Ball, 2009. ''Shapes'', pp. 231–252.</ref>
  
 
Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.
 
Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.
  
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动物标记、动物分节和叶序等生物学斑图是以不同的方式形成的。
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动物标记、动物分节和叶序等生物学斑图是以不同的方式形成的。<ref name=":3" />
  
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In [[developmental biology]], pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an [[embryo]] assume complex forms and functions.<ref>Ball, 2009. Shapes, pp. 261–290.</ref> [[Embryogenesis]], such as [[Drosophila embryogenesis|of the fruit fly ''Drosophila'']], involves coordinated [[cell fate determination|control of cell fates]].<ref name="Lai">{{cite journal |author=Eric C. Lai |title=Notch signaling: control of cell communication and cell fate |doi=10.1242/dev.01074 |pmid=14973298 |volume=131 |issue=5 |date=March 2004 |pages=965–73 |journal=Development|doi-access=free }}</ref><ref name="Tyler">{{cite journal |title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition |authors=Melinda J. Tyler, David A. Cameron|journal=Vision Research |volume=47 |issue=4 |pages=501–511 |year=2007 |doi=10.1016/j.visres.2006.08.025 |pmid=17034830}}</ref><ref name="Meinhard">{{cite web|title=Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation |author=Hans Meinhard |agency= Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany |url=http://www1.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm |date=2001-10-26 }}</ref> Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a [[morphogen]] gradient, followed by short distance cell-to-cell communication through [[cell signaling]] pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the [[French flag model]] in the 1960s.<ref>{{cite journal |doi=10.1016/S0022-5193(69)80016-0 |author=Wolpert L |title=Positional information and the spatial pattern of cellular differentiation |journal=J. Theor. Biol. |volume=25 |issue=1 |pages=1–47 |date=October 1969 |pmid=4390734 }}</ref><ref>{{cite book |author=Wolpert, Lewis |title=Principles of development |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2007 |isbn=978-0-19-927536-6 |edition=3rd |display-authors=etal}}</ref> More generally, the morphology of organisms is patterned by the mechanisms of [[evolutionary developmental biology]], such as [[heterochrony|changing the timing]] and positioning of specific developmental events in the embryo.<ref>{{cite journal |last1=Hall |first1=B. K. |title=Evo-Devo: evolutionary developmental mechanisms |journal=International Journal of Developmental Biology |date=2003 |volume=47 |issue=7–8 |pages=491–495 |pmid=14756324}}</ref>
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In [[developmental biology]], pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an [[embryo]] assume complex forms and functions.<ref name=":4">Ball, 2009. Shapes, pp. 261–290.</ref> [[Embryogenesis]], such as [[Drosophila embryogenesis|of the fruit fly ''Drosophila'']], involves coordinated [[cell fate determination|control of cell fates]].<ref name="Lai">{{cite journal |author=Eric C. Lai |title=Notch signaling: control of cell communication and cell fate |doi=10.1242/dev.01074 |pmid=14973298 |volume=131 |issue=5 |date=March 2004 |pages=965–73 |journal=Development|doi-access=free }}</ref><ref name="Tyler">{{cite journal |title=Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition |authors=Melinda J. Tyler, David A. Cameron|journal=Vision Research |volume=47 |issue=4 |pages=501–511 |year=2007 |doi=10.1016/j.visres.2006.08.025 |pmid=17034830}}</ref><ref name="Meinhard">{{cite web|title=Biological pattern formation: How cell[s] talk with each other to achieve reproducible pattern formation |author=Hans Meinhard |agency= Max-Planck-Institut für Entwicklungsbiologie, Tübingen, Germany |url=http://www1.biologie.uni-hamburg.de/b-online/e28_1/pattern.htm |date=2001-10-26 }}</ref> Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a [[morphogen]] gradient, followed by short distance cell-to-cell communication through [[cell signaling]] pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the [[French flag model]] in the 1960s.<ref name=":5">{{cite journal |doi=10.1016/S0022-5193(69)80016-0 |author=Wolpert L |title=Positional information and the spatial pattern of cellular differentiation |journal=J. Theor. Biol. |volume=25 |issue=1 |pages=1–47 |date=October 1969 |pmid=4390734 }}</ref><ref name=":6">{{cite book |author=Wolpert, Lewis |title=Principles of development |publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=2007 |isbn=978-0-19-927536-6 |edition=3rd |display-authors=etal}}</ref> More generally, the morphology of organisms is patterned by the mechanisms of [[evolutionary developmental biology]], such as [[heterochrony|changing the timing]] and positioning of specific developmental events in the embryo.<ref>{{cite journal |last1=Hall |first1=B. K. |title=Evo-Devo: evolutionary developmental mechanisms |journal=International Journal of Developmental Biology |date=2003 |volume=47 |issue=7–8 |pages=491–495 |pmid=14756324}}</ref>
  
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In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions. Embryogenesis, such as of the fruit fly Drosophila, involves coordinated control of cell fates. Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the French flag model in the 1960s. More generally, the morphology of organisms is patterned by the mechanisms of evolutionary developmental biology, such as changing the timing and positioning of specific developmental events in the embryo.
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在发育生物学中，斑图生成描述了胚胎组织发育中最初相同的细胞逐步呈现出复杂形态和功能的机制。<ref name=":4" /> 以果蝇为例，'''<font color="#ff8000">胚胎发育 Embryogenesis</font>''' 涉及到细胞命运的协调控制。<ref name="Lai" /><ref name="Tyler" /><ref name="Meinhard" /> 斑图生成是由遗传基因所控制，通常涉及一个场中的每个细胞沿着形态发生素梯度感知和响应其位置，然后通过细胞信号通路进行短距离的细胞间通信以完善初始斑图。在此背景下，'''<font color="#32CD32">细胞场 A Field of Cells</font>''' 是指通过响应同一组位置信息线索而影响其细胞命运的一组细胞。这个概念模型最早在20世纪60年代被描述为'''法旗模型 French Flag Model'''。<ref name=":5" /><ref name=":6" /> 更一般来说，生物体的形态是由'''进化发育生物学 Evolutionary Developmental Biology'''的机制，如改变胚胎中特定发育事件的时间和位置所决定的。
  
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在发育生物学中，斑图生成描述了胚胎组织发育中最初相同的细胞逐步呈现出复杂形态和功能的机制。以果蝇为例，'''<font color="#ff8000">胚胎发育 Embryogenesis</font>''' 涉及到细胞命运的协调控制。斑图生成是由遗传基因所控制，通常涉及一个场中的每个细胞沿着形态发生素梯度感知和响应其位置，然后通过细胞信号通路进行短距离的细胞间通信以完善初始斑图。在此背景下，'''<font color="#32CD32">细胞场 A Field of Cells</font>''' 是指通过响应同一组位置信息线索而影响其细胞命运的一组细胞。这个概念模型最早在20世纪60年代被描述为'''法旗模型'''。更一般来说，生物体的形态是由进化发育生物学的机制，如改变胚胎中特定发育事件的时间和位置所决定的。
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Possible mechanisms of pattern formation in biological systems include the classical [[reaction–diffusion]] model proposed by [[Alan Turing]]<ref name=":7">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</ref> and the more recently found [[elastic instability]] mechanism which is thought to be responsible for the fold patterns on the [[cerebral cortex]] of higher animals, among other things.<ref name="Mercker">{{cite journal |last1=Mercker |first1=M |last2=Brinkmann |first2=F |last3=Marciniak-Czochra |first3=A |last4=Richter |first4=T |title=Beyond Turing: mechanochemical pattern formation in biological tissues. |journal=Biology Direct |date=4 May 2016 |volume=11 |pages=22 |doi=10.1186/s13062-016-0124-7 |pmid=27145826|pmc=4857296 }}</ref><ref name=":8">Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632</ref>
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Possible mechanisms of pattern formation in biological systems include the classical [[reaction–diffusion]] model proposed by [[Alan Turing]]<ref>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</ref> and the more recently found [[elastic instability]] mechanism which is thought to be responsible for the fold patterns on the [[cerebral cortex]] of higher animals, among other things.<ref name="Mercker">{{cite journal |last1=Mercker |first1=M |last2=Brinkmann |first2=F |last3=Marciniak-Czochra |first3=A |last4=Richter |first4=T |title=Beyond Turing: mechanochemical pattern formation in biological tissues. |journal=Biology Direct |date=4 May 2016 |volume=11 |pages=22 |doi=10.1186/s13062-016-0124-7 |pmid=27145826|pmc=4857296 }}</ref><ref>Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632</ref>
 
  
 
Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.
 
Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.
  
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生物系统中可能存在的斑图生成机制包括：'''阿兰 · 图灵 Alan Turing''' 提出的经典反应—扩散模型和最近发现的弹性不稳定机制，后者被认为是高等动物大脑皮层上褶皱模式等形成的原因。
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生物系统中可能存在的斑图生成机制包括：'''阿兰 · 图灵 Alan Turing''' 提出的经典'''反应—扩散 Reaction–Diffusion'''模型<ref name=":7" /> 和最近发现的弹性不稳定机制，后者被认为是高等动物大脑皮层上褶皱模式等形成的原因。<ref name="Mercker" /><ref name=":8" />
 
====Growth of colonies 菌落生长====
 
====Growth of colonies 菌落生长====
  
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Bacterial colonies show a [[bacterial patterns|large variety of patterns]] formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns.<ref>Ball, 2009. ''Branches'', pp. 52–59.</ref> Other organisms such as [[slime mould]]s display remarkable patterns caused by the dynamics of chemical signaling.<ref>Ball, 2009. ''Shapes'', pp. 149–151.</ref>
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Bacterial colonies show a [[bacterial patterns|large variety of patterns]] formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns.<ref name=":9">Ball, 2009. ''Branches'', pp. 52–59.</ref> Other organisms such as [[slime mould]]s display remarkable patterns caused by the dynamics of chemical signaling.<ref name=":10">Ball, 2009. ''Shapes'', pp. 149–151.</ref> Cellular embodiment (elongation and adhesion) can also have an impact on the developing patterns.<ref name=":11">Duran-Nebreda, Salva; Pla, Jordi; Vidiella, Blai; Piñero, Jordi; Conde-Pueyo, Nuria; Solé, Ricard (2021-01-15). "Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies". ''ACS Synthetic Biology''. '''10''' (2): 277–285. doi:10.1021/acssynbio.0c00318. ISSN 2161-5063.</ref>
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Bacterial colonies show a large variety of patterns formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns. Other organisms such as slime moulds display remarkable patterns caused by the dynamics of chemical signaling.
 
  
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细菌菌落在生长过程中形成的斑图种类繁多，其形状取决于生长环境。特别是包括培养基环境的严酷程度、营养物质的缺乏等在内的各种外部压力，会增强了所产生斑图的复杂性。对于其他生物，如粘液霉菌，则显示出由化学信号动力学驱动的非同寻常的斑图。
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细菌菌落在生长过程中形成的斑图种类繁多，其形状取决于生长环境。特别是包括培养基环境的严酷程度、营养物质的缺乏等在内的各种外部压力，会增强了所产生斑图的复杂性。<ref name=":9" /> 对于其他生物，如粘液霉菌，则显示出由化学信号动力学驱动的非同寻常的斑图。<ref name=":10" /> 细胞体现(延伸和粘附)也会对发育模式产生影响。<ref name=":11" />
 
====Vegetation patterns 植被斑图====
 
====Vegetation patterns 植被斑图====
  
 
{{Main|patterned vegetation}}
 
{{Main|patterned vegetation}}
  
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[[patterned vegetation|Vegetation patterns]] such as [[tiger bush]]<ref name="TigerBush">{{cite book | title=Banded vegetation patterning in arid and semiarid environments | publisher=Springer-Verlag | author=Tongway, D.J., Valentin, C. & Seghieri, J. | year=2001 | location=New York|isbn=978-1461265597}}</ref> and [[fir wave]]s<ref name="FirWave">{{cite web | url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html | title=Fir Waves: Regeneration in New England Conifer Forests | publisher=TIEE | date=22 February 2004 | accessdate=26 May 2012 | author=D'Avanzo, C.}}</ref> 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.<ref name="TigerBush" /> 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<ref name="FirWave" />. 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<ref>{{cite journal |author=Meron, E |title=Vegetation pattern formation: the mechanisms behind the forms |journal=Physics Today |volume=72 |issue=11 | pages=30-36 |year=2019 |doi=10.1063/PT.3.4340}}</ref><ref>{{cite journal |author=Meron, E |title=From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study|journal=Annual Review of Condensed Matter Physics  |volume=9 | pages=79-103  |year=2018 |doi=10.1146/annurev-conmatphys-033117-053959}}</ref>.
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[[patterned vegetation|Vegetation patterns]] such as [[tiger bush]]<ref name="TigerBush">{{cite book | title=Banded vegetation patterning in arid and semiarid environments | publisher=Springer-Verlag | author=Tongway, D.J., Valentin, C. & Seghieri, J. | year=2001 | location=New York|isbn=978-1461265597}}</ref> and [[fir wave]]s<ref name="FirWave">{{cite web | url=http://tiee.esa.org/vol/v1/figure_sets/disturb/disturb_back4.html | title=Fir Waves: Regeneration in New England Conifer Forests | publisher=TIEE | date=22 February 2004 | accessdate=26 May 2012 | author=D'Avanzo, C.}}</ref> 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.<ref name="TigerBush" /> 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<ref name="FirWave" />. 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<ref name=":12">{{cite journal |author=Meron, E |title=Vegetation pattern formation: the mechanisms behind the forms |journal=Physics Today |volume=72 |issue=11 | pages=30-36 |year=2019 |doi=10.1063/PT.3.4340}}</ref><ref name=":13">{{cite journal |author=Meron, E |title=From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study|journal=Annual Review of Condensed Matter Physics  |volume=9 | pages=79-103  |year=2018 |doi=10.1146/annurev-conmatphys-033117-053959}}</ref>.
 
[[文件:Tiger Bush Niger Corona 1965-12-31.jpg|缩略图|Tiger bush is a vegetation pattern that forms in arid conditions.虎皮纹灌木是一种在干旱条件下形成的植被模式。]]
 
[[文件:Tiger Bush Niger Corona 1965-12-31.jpg|缩略图|Tiger bush is a vegetation pattern that forms in arid conditions.虎皮纹灌木是一种在干旱条件下形成的植被模式。]]
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虎皮纹灌木和冷杉波纹等植被斑图形成的原因不同。在尼日尔等干旱国家，植物生长受到降雨的限制，虎皮纹灌木由干旱斜坡上的带状灌木组成。每一条大致水平的植被带都可以从紧靠其上方的裸露地带吸收雨水。相比之下，在受到大风摧残后的植被再生过程中，山坡上的森林会出现冷杉波纹。当较外层树木倒下时，曾经受到它们庇护的内层树木会暴露在外，进而这部分植被更容易受到破坏。因此，树林空隙往往向背风面扩大。同时在迎风坡，幼树在生长过程中则受到存活下来的高大树木自然形成的风影区保护。因此在平坦的地形中，植被除了条带状之外，还出现了其他的斑图形态—例如六边形缝隙斑图和六边形点状斑图。在这种情况下，植被斑图是在当地植被生长和朝向生长位置的水分输送之间的正反馈回路驱动下形成的。
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虎皮纹灌木<ref name="TigerBush" /> 和冷杉波纹<ref name="FirWave" /> 等植被斑图形成的原因不同。在尼日尔等干旱国家，植物生长受到降雨的限制，虎皮纹灌木由干旱斜坡上的带状灌木组成。每一条大致水平的植被带都可以从紧靠其上方的裸露地带吸收雨水。<ref name="TigerBush" /> 相比之下，在受到大风摧残后的植被再生过程中，山坡上的森林会出现冷杉波纹。当较外层树木倒下时，曾经受到它们庇护的内层树木会暴露在外，进而这部分植被更容易受到破坏。因此，树林空隙往往向背风面扩大。同时在迎风坡，幼树在生长过程中则受到存活下来的高大树木自然形成的风影区保护。<ref name="FirWave" /> 因此在平坦的地形中，植被除了条带状之外，还出现了其他的斑图形态—例如六边形缝隙斑图和六边形点状斑图。在这种情况下，植被斑图是在当地植被生长和朝向生长位置的水分输送之间的正反馈回路驱动下形成的。<ref name=":12" /><ref name=":13" />
  
 
===Chemistry 化学===
 
===Chemistry 化学===
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{{expand section|date=March 2013}}
 
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{{further|reaction–diffusion system|Turing patterns}}Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.<ref name=":0">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> The [[Brusselator]] model developed by [[Ilya Prigogine]] and collaborators is one such example that exhibits [[Turing instability]].<ref>{{Citation|last=Prigogine|first=I.|title=Self-Organisation in Nonequilibrium Systems: Towards A Dynamics of Complexity|date=1985|work=Bifurcation Analysis: Principles, Applications and Synthesis|pages=3–12|editor-last=Hazewinkel|editor-first=M.|publisher=Springer Netherlands|doi=10.1007/978-94-009-6239-2_1|isbn=9789400962392|last2=Nicolis|first2=G.|editor2-last=Jurkovich|editor2-first=R.|editor3-last=Paelinck|editor3-first=J. H. P.}}</ref> Pattern formation in chemical systems often involve [[Chemical oscillator|oscillatory chemical kinetics]] or [[Autocatalysis|autocatalytic reactions]]<ref name=":1">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|issn=1934-2659}}</ref> 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]].<ref>{{Cite journal|last=Marwaha|first=Bharat|last2=Sundarram|first2=Sandhya|last3=Luss|first3=Dan|date=September 2004|title=Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†|journal=The Journal of Physical Chemistry B|volume=108|issue=38|pages=14470–14476|doi=10.1021/jp049803p|issn=1520-6106}}</ref><ref name=":0" /> The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying [[Reaction–diffusion system|reaction-diffusion system]].<ref name=":0" /><ref name=":1" />
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{{further|reaction–diffusion system|Turing patterns}}Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.<ref name=":0">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}}</ref> The [[Brusselator]] model developed by [[Ilya Prigogine]] and collaborators is one such example that exhibits [[Turing instability]].<ref name=":14">Prigogine, I.; Nicolis, G. (1985), Hazewinkel, M.; Jurkovich, R.; Paelinck, J. H. P. (eds.), "Self-Organisation in Nonequilibrium Systems: Towards A Dynamics of Complexity", ''Bifurcation Analysis: Principles, Applications and Synthesis'', Springer Netherlands, pp. 3–12, doi:10.1007/978-94-009-6239-2_1, ISBN 9789400962392</ref> Pattern formation in chemical systems often involve [[Chemical oscillator|oscillatory chemical kinetics]] or [[Autocatalysis|autocatalytic reactions]]<ref name=":1">{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|issn=1934-2659}}</ref> 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]].<ref name=":15">{{Cite journal|last=Marwaha|first=Bharat|last2=Sundarram|first2=Sandhya|last3=Luss|first3=Dan|date=September 2004|title=Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†|journal=The Journal of Physical Chemistry B|volume=108|issue=38|pages=14470–14476|doi=10.1021/jp049803p|issn=1520-6106}}</ref><ref name=":0" /> The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying [[Reaction–diffusion system|reaction-diffusion system]].<ref name=":0" /><ref name=":1" />
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Pattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns. The Brusselator model developed by Ilya Prigogine and collaborators is one such example that exhibits Turing instability. Pattern formation in chemical systems often involve oscillatory chemical kinetics or autocatalytic reactions 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. The emergence of pattern formation can be studied by mathematical modeling and simulation of the underlying reaction-diffusion system.
 
  
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在化学和化学工程领域，斑图生成的研究进展良好，其中包括温度和浓度斑图。由'''伊利亚·普利高津 Ilya Prigogine''' 和其合作者开发的布鲁塞尔器模型就是一个展示出图灵不稳定性的例子。化学体系中的斑图生成通常涉及'''<font color="#ff8000"> 振荡化学动力学 Oscillatory Chemical Kinetics</font>''' 或'''<font color="#ff8000"> 自催化反应 Autocatalytic Reactions</font>'''，如'''别洛乌索夫-扎波茨基反应Belousov–Zhabotinsky Reaction''' 或 '''Briggs–Rauscher Reaction 布里格斯-劳舍反应'''。在工业应用中，如化学反应堆，斑图生成可能导致温度热点，进而会降低产量或造成灾害性安全问题，如热失控。斑图生成的出现可以用底层的反应—扩散系统的数学建模与模拟来研究。
+
在化学和化学工程领域，斑图生成的研究进展良好，其中包括温度和浓度斑图。<ref name=":0" />由'''伊利亚·普利高津 Ilya Prigogine'''和其合作者开发的'''布鲁塞尔器 Brusselator'''模型就是一个展示出'''图灵不稳定性 Turing Instability'''的例子。<ref name=":14" /> 化学体系中的斑图生成通常涉及'''<font color="#ff8000">振荡化学动力学 Oscillatory Chemical Kinetics</font>''' 或'''<font color="#ff8000"> 自催化反应 Autocatalytic Reactions</font>'''，<ref name=":1" /> 如'''别洛乌索夫-扎波茨基反应 Belousov–Zhabotinsky Reaction'''或'''Briggs–Rauscher Reaction 布里格斯-劳舍反应'''。在工业应用中，如化学反应堆，斑图生成可能导致温度热点，进而会降低产量或造成灾害性安全问题，如热失控。<ref name=":0" /><ref name=":15" /> 斑图生成的出现可以用底层的反应—扩散系统的数学建模与模拟来研究。<ref name=":0" /><ref name=":1" />
 
* [[Belousov–Zhabotinsky reaction]]
 
* [[Belousov–Zhabotinsky reaction]]
 
A '''Belousov–Zhabotinsky reaction''', or '''BZ reaction''', is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological<sup>[''clarification needed'']</sup> phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.
 
A '''Belousov–Zhabotinsky reaction''', or '''BZ reaction''', is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological<sup>[''clarification needed'']</sup> phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.
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===Computer graphics 计算机图形学===
 
===Computer graphics 计算机图形学===
  
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[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[reaction–diffusion]] model, produced using sharpen and blur|链接=Special:FilePath/Homebrew_reaction_diffusion_example_512iter.jpg]]
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[[File:Homebrew reaction diffusion example 512iter.jpg|thumb|right|Pattern resembling a [[reaction–diffusion]] model, produced using sharpen and blur
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利用图像锐化和虚化产生的斑图近似反应-扩散模型|链接=Special:FilePath/Homebrew_reaction_diffusion_example_512iter.jpg]]
  
  

2021年8月1日 (日) 16:37的版本

本词条由11初步翻译，由CecileLi初步审校 https://wiki.swarma.org/index.php?title=%E5%B9%B3%E8%A1%A1%E7%90%86%E8%AE%BA#:~:text=%E6%9C%AC%E8%AF%8D%E6%9D%A1%E7%94%B1,11%E5%88%9D%E6%AD%A5%E7%BF%BB%E8%AF%91由栗子CUGB继续完善

Complex systems organizational map.jpg
Complex systems
Topics
文件:Self-organizing-Mechanism-for-Development-of-Space-filling-Neuronal-Dendrites-pcbi.0030212.sv003.ogv
Pattern formation in a computational model of dendrite growth.
文件:用于开发填充神经网络树突神经的自发性组织机制-pcbi.0030212.sv003.ogv
Pattern formation in a computational model of dendrite growth.

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.

The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature.

斑图生成学关注自组织中可见的、统计有序的结果，以及自然界中相似斑图背后的共同原理。

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. 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.

In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. Pattern formation is controlled by genes. 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.

在 发育生物学 Developmental Biology 中，斑图生成指的是细胞命运中复杂组织的产生过程。斑图生成由基因控制的。基因在斑图生成中所起的作用属于形态发生的一个方面，即由相似的基因演化出不同的生命结构，当下也属于演化发育生物学所探究的问题。模式生物黑腹果蝇（第一个应用到形态发生研究的物种）的胚胎的斑图生成过程和蝴蝶的眼点清楚地体现了这一机制，而后者的发育过程是标准（果蝇）机制的一种变体。

Examples 实例

模板:Further 模板:进一步

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

Examples of pattern formation can be found in biology, chemistry, physics, and mathematics, and can readily be simulated with computer graphics, as described in turn below.

在生物、化学、物理和数学[1] 中都有斑图生成的实例，并且我们可以用 计算机图形学 Computer Graphics 轻松地来模拟，下面依次进行介绍。

Biology 生物学

模板:Further 模板:深入研究

Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.[2]

Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways.

动物标记、动物分节和叶序等生物学斑图是以不同的方式形成的。[2]

In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions.[3] Embryogenesis, such as of the fruit fly Drosophila, involves coordinated control of cell fates.[4][5][6] Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short distance cell-to-cell communication through cell signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set positional information cues. This conceptual model was first described as the French flag model in the 1960s.[7][8] More generally, the morphology of organisms is patterned by the mechanisms of evolutionary developmental biology, such as changing the timing and positioning of specific developmental events in the embryo.[9]

在发育生物学中，斑图生成描述了胚胎组织发育中最初相同的细胞逐步呈现出复杂形态和功能的机制。[3] 以果蝇为例，胚胎发育 Embryogenesis 涉及到细胞命运的协调控制。[4][5][6] 斑图生成是由遗传基因所控制，通常涉及一个场中的每个细胞沿着形态发生素梯度感知和响应其位置，然后通过细胞信号通路进行短距离的细胞间通信以完善初始斑图。在此背景下，细胞场 A Field of Cells 是指通过响应同一组位置信息线索而影响其细胞命运的一组细胞。这个概念模型最早在20世纪60年代被描述为法旗模型 French Flag Model。[7][8] 更一般来说，生物体的形态是由进化发育生物学 Evolutionary Developmental Biology的机制，如改变胚胎中特定发育事件的时间和位置所决定的。

Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing[10] and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.[11][12]

Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more recently found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things.

生物系统中可能存在的斑图生成机制包括：阿兰 · 图灵 Alan Turing 提出的经典反应—扩散 Reaction–Diffusion模型[10] 和最近发现的弹性不稳定机制，后者被认为是高等动物大脑皮层上褶皱模式等形成的原因。[11][12]

Growth of colonies 菌落生长

Bacterial colonies show a large variety of patterns formed during colony growth. The resulting shapes depend on the growth conditions. In particular, stresses (hardness of the culture medium, lack of nutrients, etc.) enhance the complexity of the resulting patterns.[13] Other organisms such as slime moulds display remarkable patterns caused by the dynamics of chemical signaling.[14] Cellular embodiment (elongation and adhesion) can also have an impact on the developing patterns.[15]

细菌菌落在生长过程中形成的斑图种类繁多，其形状取决于生长环境。特别是包括培养基环境的严酷程度、营养物质的缺乏等在内的各种外部压力，会增强了所产生斑图的复杂性。[13] 对于其他生物，如粘液霉菌，则显示出由化学信号动力学驱动的非同寻常的斑图。[14] 细胞体现(延伸和粘附)也会对发育模式产生影响。[15]

Vegetation patterns 植被斑图

Main article: patterned vegetation

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].

Tiger bush is a vegetation pattern that forms in arid conditions.虎皮纹灌木是一种在干旱条件下形成的植被模式。

虎皮纹灌木[16] 和冷杉波纹[17] 等植被斑图形成的原因不同。在尼日尔等干旱国家，植物生长受到降雨的限制，虎皮纹灌木由干旱斜坡上的带状灌木组成。每一条大致水平的植被带都可以从紧靠其上方的裸露地带吸收雨水。[16] 相比之下，在受到大风摧残后的植被再生过程中，山坡上的森林会出现冷杉波纹。当较外层树木倒下时，曾经受到它们庇护的内层树木会暴露在外，进而这部分植被更容易受到破坏。因此，树林空隙往往向背风面扩大。同时在迎风坡，幼树在生长过程中则受到存活下来的高大树木自然形成的风影区保护。[17] 因此在平坦的地形中，植被除了条带状之外，还出现了其他的斑图形态—例如六边形缝隙斑图和六边形点状斑图。在这种情况下，植被斑图是在当地植被生长和朝向生长位置的水分输送之间的正反馈回路驱动下形成的。[18][19]

Chemistry 化学

模板:Expand section

模板:FurtherPattern formation has been well studied in chemistry and chemical engineering, including both temperature and concentration patterns.[20] The Brusselator model developed by Ilya Prigogine and collaborators is one such example that exhibits 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]

在化学和化学工程领域，斑图生成的研究进展良好，其中包括温度和浓度斑图。[20]由伊利亚·普利高津 Ilya Prigogine和其合作者开发的布鲁塞尔器 Brusselator模型就是一个展示出图灵不稳定性 Turing Instability的例子。[21] 化学体系中的斑图生成通常涉及振荡化学动力学 Oscillatory Chemical Kinetics 或 自催化反应 Autocatalytic Reactions，[22] 如别洛乌索夫-扎波茨基反应 Belousov–Zhabotinsky Reaction或Briggs–Rauscher Reaction 布里格斯-劳舍反应。在工业应用中，如化学反应堆，斑图生成可能导致温度热点，进而会降低产量或造成灾害性安全问题，如热失控。[20][23] 斑图生成的出现可以用底层的反应—扩散系统的数学建模与模拟来研究。[20][22]

A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological[clarification needed] phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order.

别洛乌索夫-扎波茨基反应(BZ 反应) 是一类作为非平衡热力学经典例子的反应，会促成非线性化学振荡器的建立。（这些振子中唯一共同的元素是溴和酸。）这类反应对于理论化学很重要，因为它们表明化学反应不必受平衡热力学行为的支配。这些反应远未达到平衡，并在相当长的一段时间内保持非平衡状态，并无序地发展。从这个意义上说，它们为非平衡的生物学现象(需要澄清？)提供了一个有价值的化学模型；因此，BZ反应本身的数学模型和模拟具有理论意义，揭示某类现象作为噪声驱动的秩序的性质。

An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quiescent medium. Some clock reactions such as Briggs–Rauscher and BZ using the tris(bipyridine)ruthenium(II) chloride as catalyst can be excited into self-organising activity through the influence of light.

BZ反应的一个本质内容就是所谓的“可激发性”；在刺激因素的影响下，斑图可以在原本完全静止的介质中形成。一些时钟反应，如Briggs-Rauscher和BZ，使用三(联吡啶)氯化钌作为催化剂，可以在光的影响下激发产生自组织活动。

Physics 物理学

模板:Expand section

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.

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.

20世纪80年代，卢贾托 Lugiato和勒弗 Lefever开发了一个光在光学谐振腔中传播的模型，该模型通过利用非线性效应形成斑图。

Bénard cells, laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. Dendrites in solidification, liquid crystals. Solitons.

Bénard cells, laser, cloud formations in stripes or rolls. Ripples in icicles. Washboard patterns on dirtroads. Dendrites in solidification, liquid crystals. Solitons.

如伯纳德涡流、激光、条状云或卷状云、冰柱上的涟漪、泥路上的洗衣板等图案以及凝固中的树枝状、液晶、孤子等。

Rayleigh-Bénard convection is a type of natural convection, occurring in a planar horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Bénard–Rayleigh convection is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems.

瑞利-伯纳德对流是一种自然对流，当从下方加热流体时，出现在流体表面水平层中，其中流体会产生一种形似细胞的规则对流斑图，称为伯纳德涡流。由于无论是进行理论分析还是实验研究都很方便，瑞利-伯纳德对流是研究最广泛的对流现象之一。这一对流斑图是自组织非线性系统中研究最透彻的一例。

Buoyancy, and hence gravity, are responsible for the appearance of convection cells. The initial movement is the upwelling of lesser density fluid from the heated bottom layer. This upwelling spontaneously organizes into a regular pattern of cells.

浮力和重力的影响是对流细胞斑图出现的原因。最初的运动是密度较小的流体从受热的底层上涌。这种上升流自发地组织成一种规则的细胞斑图。

In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

在数学和物理学中，孤子或孤立波是一种自我增强的波包，在以恒定速度传播时保持其形状。孤子是由于介质中非线性和色散效应的抵消而产生的。（色散效应是某些系统的一种特性，在这种系统中，波的速度取决于它的频率。）孤子是描述物理系统的一类广泛存在的弱非线性色散偏微分方程的解。

The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

1834年，约翰·斯科特·罗素 John Scott Russell（1808-1882）在苏格兰联合运河观测到一个孤波，首次描述了孤波现象。之后他在一个波浪槽中再现了这一现象，并将其命名为“转化波”。

Mathematics 数学

模板:Expand section

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

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

例如最密堆积（球填充）和覆盖。数学是其他斑图生成机制的基础。

In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.

在几何学中，球填充是在一个包含空间内不重叠的球的排列。球通常都具有相同的尺寸，填充空间一般指三维的欧几里得空间。然而，球填充问题可以推广到不相等的球、其他维度的空间(二维的问题变成圆填充，高维的问题变成超球填充)或非欧几里得空间，如双曲空间。

A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.

一个典型的球填充问题是找到一种安排，在其中球填充尽可能多的空间。球体所占空间的比例称为排列密度。由于在无限空间中填充的局部密度可能随测量的体积而变化，因此问题通常是使在足够大的体积上测量的平均密度或渐近密度最大化。

For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.

对于三维尺寸相等的球体，最密的填充使用大约74%的体积。随机排列的等量球体的密度通常在64%左右。

模板:Further 模板:进一步

Computer graphics 计算机图形学

Pattern resembling a reaction–diffusion model, produced using sharpen and blur 利用图像锐化和虚化产生的斑图近似反应-扩散模型


模板:Further 模板:进一步

Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects.[24][25]

Some types of automata have been used to generate organic-looking textures for more realistic shading of 3d objects.

一些类型的自动机已经用来生成类似有机外观的纹理，为三维物体提供更真实的阴影效果。

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

KPT6是流行的Photoshop插件，，包一个名为“KPT反应”的过滤器。反应该根据提供的种子图像产生反应-扩散风格的图案。

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.

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.

在数字图像处理中，只要有一点耐心，通过在图形编辑器中反复锐化和模糊图像，就可以用卷积函数实现类似于“KPT反应”的效果。如果使用其他滤波器，如浮雕或边缘检测，可以实现不同类型的效果。

Computers are often used to simulate the biological, physical or chemical processes that lead to pattern formation, and they can display the results in a realistic way. Calculations using models like reaction–diffusion or MClone are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

Computers are often used to simulate the biological, physical or chemical processes that lead to pattern formation, and they can display the results in a realistic way. Calculations using models like reaction–diffusion or MClone are based on the actual mathematical equations designed by the scientists to model the studied phenomena.

计算机经常被用来模拟导致斑图生成的生物、物理或化学过程，它们可以以逼真的方式显示结果。使用反应—扩散或MClone等模型的计算是基于科学家们为模拟研究现象而设计的实际数学方程。

References

参考资料

  1. ↑ 1.0 1.1 Ball, 2009.
  2. ↑ 2.0 2.1 Ball, 2009. Shapes, pp. 231–252.
  3. ↑ 3.0 3.1 Ball, 2009. Shapes, pp. 261–290.
  4. ↑ 4.0 4.1 Eric C. Lai (March 2004). "Notch signaling: control of cell communication and cell fate". Development. 131 (5): 965–73. doi:10.1242/dev.01074. PMID 14973298.
  5. ↑ 5.0 5.1 Melinda J. Tyler, David A. Cameron (2007). "Cellular pattern formation during retinal regeneration: A role for homotypic control of cell fate acquisition". Vision Research. 47 (4): 501–511. doi:10.1016/j.visres.2006.08.025. PMID 17034830.{{cite journal}}: CS1 maint: uses authors parameter (link)
  6. ↑ 6.0 6.1 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. ↑ 7.0 7.1 Wolpert L (October 1969). "Positional information and the spatial pattern of cellular differentiation". J. Theor. Biol. 25 (1): 1–47. doi:10.1016/S0022-5193(69)80016-0. PMID 4390734.
  8. ↑ 8.0 8.1 Wolpert, Lewis (2007). Principles of development (3rd ed.). Oxford [Oxfordshire]: Oxford University Press. ISBN 978-0-19-927536-6. 
  9. ↑ Hall, B. K. (2003). "Evo-Devo: evolutionary developmental mechanisms". International Journal of Developmental Biology. 47 (7–8): 491–495. PMID 14756324.
  10. ↑ 10.0 10.1 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. ↑ 11.0 11.1 Mercker, M; Brinkmann, F; Marciniak-Czochra, A; Richter, T (4 May 2016). "Beyond Turing: mechanochemical pattern formation in biological tissues". Biology Direct. 11: 22. doi:10.1186/s13062-016-0124-7. PMC 4857296. PMID 27145826.
  12. ↑ 12.0 12.1 Tallinen et al. Nature Physics 12, 588–593 (2016) doi:10.1038/nphys3632
  13. ↑ 13.0 13.1 Ball, 2009. Branches, pp. 52–59.
  14. ↑ 14.0 14.1 Ball, 2009. Shapes, pp. 149–151.
  15. ↑ 15.0 15.1 Duran-Nebreda, Salva; Pla, Jordi; Vidiella, Blai; Piñero, Jordi; Conde-Pueyo, Nuria; Solé, Ricard (2021-01-15). "Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies". ACS Synthetic Biology. 10 (2): 277–285. doi:10.1021/acssynbio.0c00318. ISSN 2161-5063.
  16. ↑ 16.0 16.1 16.2 16.3 Tongway, D.J., Valentin, C. & Seghieri, J. (2001). Banded vegetation patterning in arid and semiarid environments. New York: Springer-Verlag. ISBN 978-1461265597. 
  17. ↑ 17.0 17.1 17.2 17.3 D'Avanzo, C. (22 February 2004). "Fir Waves: Regeneration in New England Conifer Forests". TIEE. Retrieved 26 May 2012.
  18. ↑ 18.0 18.1 Meron, E (2019). "Vegetation pattern formation: the mechanisms behind the forms". Physics Today. 72 (11): 30–36. doi:10.1063/PT.3.4340.
  19. ↑ 19.0 19.1 Meron, E (2018). "From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study". Annual Review of Condensed Matter Physics. 9: 79–103. doi:10.1146/annurev-conmatphys-033117-053959.
  20. ↑ 20.0 20.1 20.2 20.3 20.4 20.5 Gupta, Ankur; Chakraborty, Saikat (January 2009). "Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors". Chemical Engineering Journal. 145 (3): 399–411. doi:10.1016/j.cej.2008.08.025. ISSN 1385-8947.
  21. ↑ 21.0 21.1 Prigogine, I.; Nicolis, G. (1985), Hazewinkel, M.; Jurkovich, R.; Paelinck, J. H. P. (eds.), "Self-Organisation in Nonequilibrium Systems: Towards A Dynamics of Complexity", Bifurcation Analysis: Principles, Applications and Synthesis, Springer Netherlands, pp. 3–12, doi:10.1007/978-94-009-6239-2_1, ISBN 9789400962392
  22. ↑ 22.0 22.1 22.2 22.3 Gupta, Ankur; Chakraborty, Saikat (2008-01-19). "Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions". Chemical Product and Process Modeling. 3 (2). doi:10.2202/1934-2659.1135. ISSN 1934-2659.
  23. ↑ 23.0 23.1 Marwaha, Bharat; Sundarram, Sandhya; Luss, Dan (September 2004). "Dynamics of Transversal Hot Zones in Shallow Packed-Bed Reactors†". The Journal of Physical Chemistry B. 108 (38): 14470–14476. doi:10.1021/jp049803p. ISSN 1520-6106.
  24. ↑ Greg Turk, Reaction–Diffusion
  25. ↑ Andrew Witkin; Michael Kassy (1991). "Reaction–Diffusion Textures" (PDF). Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques: 299–308. doi:10.1145/122718.122750. ISBN 0897914368.

Bibliography

参考书目

External links 外部链接

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

模板:Patterns in nature

模板:Genarch

Category:Developmental biology

类别: 发育生物学

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Category:Pattern formation

类别: 模式形成


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