text:
First Amendment of the Constitution of India
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Part of a series on
Constitution of India
Preamble
Parts[show]
Fundamental Rights
PART
I ∙ II ∙ III ∙ IV ∙ IVA ∙ V ∙ VI ∙ VII
VIII ∙ IX ∙ IXA ∙ IXB ∙ X ∙ XI ∙ XII ∙ XIII ∙ XIV
XV ∙ XVI ∙ XVIA ∙ XVII ∙ XVIII ∙ XIX ∙ XX ∙ XXI
XXII
Schedules[show]
First ∙ Second ∙ Third ∙ Fourth ∙ Fifth
Sixth ∙ Seventh ∙ Eighth ∙ Ninth
Tenth ∙ Eleventh ∙ Twelfth
Appendices[show]
I ∙ II ∙ III ∙ IV ∙ V
Amendments[show]
List ∙ 1 ∙ 2 ∙ 3 ∙ 4 ∙ 5 ∙ 6 ∙ 7 ∙ 8 ∙ 9 ∙ 10 ∙ 11 ∙ 12 ∙ 13 ∙ 14 ∙ 15 ∙ 16 ∙ 17 ∙ 18 ∙ 19 ∙ 20 ∙ 21 ∙ 22 ∙ 23 ∙ 24 ∙ 25 ∙ 26 ∙ 27 ∙ 28 ∙ 29 ∙ 30 ∙ 31 ∙ 32 ∙ 33 ∙ 34 ∙ 35 ∙ 36 ∙ 37 ∙ 38 ∙ 39 ∙ 40 ∙ 41 ∙ 42 ∙ 43 ∙ 44 ∙ 45 ∙ 46 ∙ 47 ∙ 48 ∙ 49 ∙ 50 ∙ 51 ∙ 52 ∙ 53 ∙ 54 ∙ 55 ∙ 56 ∙ 57 ∙ 58 ∙ 59 ∙ 60 ∙ 61 ∙ 62 ∙ 63 ∙ 64 ∙ 65 ∙ 66 ∙ 67 ∙ 68 ∙ 69 ∙ 70 ∙ 71 ∙ 72 ∙ 73 ∙ 74 ∙ 75 ∙ 76 ∙ 77 ∙ 78 ∙ 79 ∙ 80 ∙ 81 ∙ 82 ∙ 83 ∙ 84 ∙ 85 ∙ 86 ∙ 87 ∙ 88 ∙ 89 ∙ 90 ∙ 91 ∙ 92 ∙ 93 ∙ 94 ∙ 95 ∙ 96 ∙ 97 ∙ 98 ∙ 99 ∙ 100 ∙ 101
Related topics[show]
Union List
State List
Concurrent List
Basic structure doctrine
Uniform civil code
v
t
e
The First Amendment of the Constitution of India, enacted in 1951, made several changes to the Fundamental Rights provisions of the constitution. It provided against abuse of freedom of speech and expression, validation of zamindari abolition laws, and clarified that the right to equality does not bar the enactment of laws which provide "special consideration" for weaker sections of society.
The formal title of the amendment is the Constitution (First Amendment) Act, 1951. It was moved by the then Prime Minister of India, Jawaharlal Nehru, on 10 May 1951 and enacted by Parliament on 18 June 1951.[1]
This Amendment set the precedent of amending the Constitution to overcome judicial judgements impeding fulfilment of the government's perceived responsibilities to particular policies and programmes.
Contents
[hide]
1 Freedom of speech
2 Freedom of trade
3 Upholding land reforms
4 Equality
4.1 Background
5 Other amendments
6 References
7 External links
Freedom of speech[edit]
In 1950, a leftist weekly journal in English, Cross Roads, published by Romesh Thapar was banned by the Madras State for publishing critical views on Nehruvian policy, who petitioned the Supreme Court, which led to the landmark judgment in "Romesh Thappar vs The State Of Madras" on 26 May 1950. Eventually, in 1951 Nehru administration made the Amendment to 19(1)(a) of Constitution of India against "abuse of freedom of speech and expression".[2][3]
Some courts had held the citizen's right to freedom of speech and expression guaranteed by Article 19(1)(a) of the Constitution of India to be so comprehensive as not to render a person culpable even if he advocates murder and other crimes of violence. The Parliament of India noted that in other countries with written constitutions, freedom of speech and of the press is not regarded as debarring the State from punishing or preventing abuse of this freedom.[1]
Freedom of trade[edit]
The right of citizens of India to practise any profession or to carry on any occupation, trade or business conferred by article 19(1)(g) is subject to reasonable restrictions which the laws of the State may impose "in the interests of general public". While the words cited are comprehensive enough to cover any scheme of nationalisation, it was thought desirable to place the matter beyond doubt by a clarificatory addition to article 19(6).[1]
Upholding land reforms[edit]
The Parliament of India noted that validity of agrarian reform measures passed by the State Legislatures had, in spite of the provisions of clauses (4) and (6) of article 31, formed the subject-matter of dilatory litigation, as a result of which the implementation of these important measures, affecting large numbers of people, had been held up. Accordingly, a new article 31A was introduced with retrospective effect to uphold such measures. Further, another new article 31B was introduced to validate 13 enactments relating to zamindari abolition.[1]
Equality[edit]
It is laid down in Article 46 as a Directive Principle of State Policy that the State should promote with special care the educational and economic interests of the weaker sections of the people and protect them from social injustice. In order that any special provision that the State may make for the educational, economic or social advancement of any backward class of citizens may not be challenged on the ground of being discriminatory, article 15(3) was suitably amplified.[1]
Background[edit]
Jawaharlal Nehru encouraged the Parliament of India to pass the amendment in response to State of Madras v. Champakam Dorairajan, which went before the Madras High Court and then the Supreme Court of India. In that case, a Brahmin woman in Madras challenged the state's Communal General Order, which established caste quotas in government-supported medical and engineering schools, on the grounds that it denied her equality under the law; both courts had upheld her petition.[4] Aside from Nehru, B. R. Ambedkar also spoke in support of the proposed amendment, while Syama Prasad Mookerjee spoke in opposition to it.[5]
Other amendments[edit]
Certain amendments in respect of articles dealing with the convening and proroguing of the sessions of Parliament were also incorporated in the Act. So also a few minor amendments in respect of articles 341, 342, 372, and 376.[1]
References[edit]
^ Jump up to: a b c d e f Full text of First Amendment
Jump up ^ "Romesh Thappar vs The State Of Madras on 26 May, 1950". indiankanoon.org. Retrieved 2014-01-14.
Jump up ^ "Half a century of ideas". Indian Express. Oct 25, 2009. Retrieved 2014-01-14.
Jump up ^ Hasan, Zoya; Sridharan, Eswaran; Sudarshan, R. (2005), India's living constitution: ideas, practices, controversies, Anthem South Asian studies, Anthem Press, p. 321, ISBN 978-1-84331-137-9
Jump up ^ Khanna, Hans Raj (2008), Making of India's Constitution (2nd ed.), Eastern Book Company, p. 224, ISBN 978-81-7012-108-4
External links[edit]
Full text from India's Ministry of Law and Justice
Retrieved from "https://en.wikipedia.org/w/index.php?title=First_Amendment_of_the_Constitution_of_India&oldid=820211572"
Categories:
Amendments of the Constitution of India
1951 in India
1951 in law
Nehru administration
Freedom of expression in India
Hidden categories:
Use dmy dates from February 2014
Use Indian English from February 2014
All Wikipedia articles written in Indian English
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
हिन्दी
Edit links
This page was last edited on 13 January 2018, at 18:02.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

key: html

type: local

value: div

attrs: id class

values: mw-page-base noprint

values: mw-head-base noprint

value: a

attrs: id class lang

values: firstHeading firstHeading en

values: siteSub noprint

values: jump-to-nav mw-jump

attrs: href title

value: td

value: tr

value: th

value: img

attrs: alt src width height srcset data-file-width data-file-height

attrs: href class

value: b

attrs: class style

values:
plainlist
padding:0 0.1em 0.4em;padding-top:0;

attrs: class id href

values: NavToggle NavToggle1 #

values:
/wiki/Fundamental_Rights_in_India
mw-redirect
Fundamental Rights in India

values: Part Five of the Constitution of India (page does not exist)

values: Part Six of the Constitution of India (page does not exist)

values: Part Seven of the Constitution of India (page does not exist)

values: Part Eight of the Constitution of India (page does not exist)

values: Part IX of the Constitution of India (page does not exist)

values: Part IXA of the Constitution of India (page does not exist)

values: Part IXB of the Constitution of India (page does not exist)

values: Part Eleven of the Constitution of India

values: Part Twelve of the Constitution of India

values: Part Thirteen of the Constitution of India

values: Part Fourteen of the Constitution of India

values: Part Fifteen of the Constitution of India

values: Part XVIA of the Constitution of India (page does not exist)

values: Part Seventeen of the Constitution of India

values: Part Eighteen of the Constitution of India

values: Part Nineteen of the Constitution of India

values: Part Twenty of the Constitution of India

values: Part Twenty-one of the Constitution of India

values: Part Twenty-two of the Constitution of India

values: font-size: 105%; padding: 0.2em 0px 0.4em; text-align: center; display: none;

values: First Schedule to the Constitution of India (page does not exist)

values: Second Schedule to the Constitution of India (page does not exist)

values: Third Schedule to the Constitution of India (page does not exist)

values: Fourth Schedule to the Constitution of India

values: Fifth Schedule to the Constitution of India (page does not exist)

values: Sixth Schedule to the Constitution of India (page does not exist)

values: Seventh Schedule to the Constitution of India (page does not exist)

values: Eighth Schedule to the Constitution of India

values: Ninth Schedule to the Constitution of India (page does not exist)

values: Tenth Schedule to the Constitution of India (page does not exist)

values: Eleventh Schedule to the Constitution of India (page does not exist)

values: Twelfth Schedule to the Constitution of India (page does not exist)

values: Appendix One of the Constitution of India (page does not exist)

values: Appendix Two of the Constitution of India (page does not exist)

values: Appendix Three of the Constitution of India (page does not exist)

values: Appendix Four of the Constitution of India (page does not exist)

values: Appendix Five of the Constitution of India (page does not exist)

values: List of amendments of the Constitution of India

values: Second Amendment of the Constitution of India

values: Third Amendment of the Constitution of India

values: Fourth Amendment of the Constitution of India (page does not exist)

values: Fifth Amendment of the Constitution of India

values: Sixth Amendment of the Constitution of India

values: Seventh Amendment of the Constitution of India (page does not exist)

values: Eighth Amendment of the Constitution of India

values: Ninth Amendment of the Constitution of India (page does not exist)

values: Tenth Amendment of the Constitution of India

values: Eleventh Amendment of the Constitution of India

values: Twelfth Amendment of the Constitution of India

values: Thirteenth Amendment of the Constitution of India (page does not exist)

values: Fourteenth Amendment of the Constitution of India

values: Fifteenth Amendment of the Constitution of India (page does not exist)

values: Sixteenth Amendment of the Constitution of India (page does not exist)

values: Seventeenth Amendment of the Constitution of India (page does not exist)

values: Eighteenth Amendment of the Constitution of India

values: Nineteenth Amendment of the Constitution of India

values: Twentieth Amendment of the Constitution of India

values: Twenty-first Amendment of the Constitution of India

values: Twenty-second Amendment of the Constitution of India

values: Twenty-third Amendment of the Constitution of India

values: Twenty-fourth Amendment of the Constitution of India

values: Twenty-fifth Amendment of the Constitution of India

values: Twenty-sixth Amendment of the Constitution of India

values: Twenty-seventh Amendment of the Constitution of India (page does not exist)

values: Twenty-eighth Amendment of the Constitution of India (page does not exist)

values: Twenty-ninth Amendment of the Constitution of India (page does not exist)

values: Thirtieth Amendment of the Constitution of India (page does not exist)

values: Thirty-first Amendment of the Constitution of India (page does not exist)

values: Thirty-second Amendment of the Constitution of India (page does not exist)

values: Thirty-third Amendment of the Constitution of India (page does not exist)

values: Thirty-fourth Amendment of the Constitution of India (page does not exist)

values: Thirty-fifth Amendment of the Constitution of India (page does not exist)

values: Thirty-sixth Amendment of the Constitution of India (page does not exist)

values: Thirty-seventh Amendment of the Constitution of India (page does not exist)

values: Thirty-eighth Amendment of the Constitution of India

values: Thirty-ninth Amendment of the Constitution of India

values: Fortieth Amendment of the Constitution of India (page does not exist)

values: Forty-first Amendment of the Constitution of India

values: Forty-second Amendment of the Constitution of India

values: Forty-third Amendment of the Constitution of India (page does not exist)

values: Forty-fourth Amendment of the Constitution of India (page does not exist)

values: Forty-fifth Amendment of the Constitution of India

values: Forty-sixth Amendment of the Constitution of India (page does not exist)

values: Forty-seventh Amendment of the Constitution of India

values: Forty-eighth Amendment of the Constitution of India

values: Forty-ninth Amendment of the Constitution of India (page does not exist)

values: Fiftieth Amendment of the Constitution of India (page does not exist)

values: Fifty-first Amendment of the Constitution of India (page does not exist)

values: Fifty-second Amendment of the Constitution of India (page does not exist)

values: Fifty-third Amendment of the Constitution of India (page does not exist)

values: Fifty-fourth Amendment of the Constitution of India (page does not exist)

values: Fifty-fifth Amendment of the Constitution of India (page does not exist)

values: Fifty-sixth Amendment of the Constitution of India (page does not exist)

values: Fifty-seventh Amendment of the Constitution of India (page does not exist)

values: Fifty-eighth Amendment of the Constitution of India (page does not exist)

values: Fifty-ninth Amendment of the Constitution of India (page does not exist)

values: Sixtieth Amendment of the Constitution of India

values: Sixty-first Amendment of the Constitution of India

values: Sixty-second Amendment of the Constitution of India

values: Sixty-third Amendment of the Constitution of India (page does not exist)

values: Sixty-fourth Amendment of the Constitution of India (page does not exist)

values: Sixty-fifth Amendment of the Constitution of India (page does not exist)

values: Sixty-sixth Amendment of the Constitution of India (page does not exist)

values: Sixty-seventh Amendment of the Constitution of India (page does not exist)

values: Sixty-eighth Amendment of the Constitution of India (page does not exist)

values: Sixty-ninth Amendment of the Constitution of India (page does not exist)

values: Seventieth Amendment of the Constitution of India (page does not exist)

values: Seventy-first Amendment of the Constitution of India

values: Seventy-second Amendment of the Constitution of India (page does not exist)

values: Seventy-third Amendment of the Constitution of India (page does not exist)

values: Seventy-fourth Amendment of the Constitution of India (page does not exist)

values: Seventy-fifth Amendment of the Constitution of India (page does not exist)

values: Seventy-sixth Amendment of the Constitution of India (page does not exist)

values: Seventy-seventh Amendment of the Constitution of India (page does not exist)

values: Seventy-eighth Amendment of the Constitution of India (page does not exist)

values: Seventy-ninth Amendment of the Constitution of India

values: Eightieth Amendment of the Constitution of India (page does not exist)

values: Eighty-first Amendment of the Constitution of India (page does not exist)

values: Eighty-second Amendment of the Constitution of India (page does not exist)

values: Eighty-third Amendment of the Constitution of India (page does not exist)

values: Eighty-fourth Amendment of the Constitution of India (page does not exist)

values: Eighty-fifth Amendment of the Constitution of India (page does not exist)

values: Eighty-sixth Amendment of the Constitution of India (page does not exist)

values: Eighty-seventh Amendment of the Constitution of India (page does not exist)

values: Eighty-eighth Amendment of the Constitution of India (page does not exist)

values: Eighty-ninth Amendment of the Constitution of India (page does not exist)

values: Ninetieth Amendment of the Constitution of India (page does not exist)

values: Ninety-first Amendment of the Constitution of India (page does not exist)

values: Ninety-second Amendment of the Constitution of India

values: Ninety-third Amendment of the Constitution of India (page does not exist)

values: Ninety-fourth Amendment of the Constitution of India

values: Ninety-fifth Amendment of the Constitution of India

values: Ninety-sixth Amendment of the Constitution of India (page does not exist)

values: Ninety-seventh Amendment of the Constitution of India (page does not exist)

values: Ninety-eighth Amendment of the Constitution of India (page does not exist)

values: Ninety-ninth Amendment of the Constitution of India (page does not exist)

values: One Hundredth Amendment of the Constitution of India (page does not exist)

values: One Hundred and First Amendment of the Constitution of India

values: float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f9f9f9;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%

values:
/wiki/Syama_Prasad_Mookerjee
mw-redirect
Syama Prasad Mookerjee

values:
nofollow
http://indiankanoon.org/doc/456839/
external text

values:
reflist mw-references-wrap
list-style-type: decimal;

values: Category:Amendments of the Constitution of India

values: Category:Use dmy dates from February 2014

values: Category:Use Indian English from February 2014

values: Category:All Wikipedia articles written in Indian English

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values: You're encouraged to log in; however, it's not mandatory. [ctrl-alt-o]

values: Discussion about the content page [ctrl-alt-t]

values: p-namespaces navigation vectorTabs p-namespaces-label

values:
p-variants
navigation
vectorMenu emptyPortlet
p-variants-label

values: Past revisions of this page [ctrl-alt-h]

values: p-views navigation vectorTabs p-views-label

values:
p-cactions
navigation
vectorMenu emptyPortlet
p-cactions-label

values:
mw-wiki-logo
/wiki/Main_Page
Visit the main page

values:
/wiki/Main_Page
Visit the main page [ctrl-alt-z]
z

values: Featured content – the best of Wikipedia

values: Find background information on current events

values:
/wiki/Special:Random
Load a random article [ctrl-alt-x]
x

values: portal navigation p-navigation p-navigation-label

values: Guidance on how to use and edit Wikipedia

values: पहला संविधान संशोधन अधिनियम, १९५१ – Hindi

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-First_Amendment_of_the_Constitution_of_India rootpage-First_Amendment_of_the_Constitution_of_India skin-vector action-view


text:
Patterns in nature
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Natural patterns form as wind blows sand in the dunes of the 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 natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.[1] Early 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, Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry. Scottish biologist D'Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. Hungarian biologist Aristid Lindenmayer and French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.
Mathematics, physics and chemistry can explain patterns in nature at different levels. Patterns in living things are explained by the biological processes of natural selection and sexual selection. Studies of pattern formation make use of computer models to simulate a wide range of patterns.
Contents
1 History
2 Causes
3 Types of pattern
3.1 Symmetry
3.2 Trees, fractals
3.3 Spirals
3.4 Chaos, flow, meanders
3.5 Waves, dunes
3.6 Bubbles, foam
3.7 Tessellations
3.8 Cracks
3.9 Spots, stripes
3.10 Pattern formation
4 See also
5 References
6 Bibliography
6.1 Pioneering authors
6.2 General books
6.3 Patterns from nature (as art)
7 External links
History[edit]
Fibonacci patterns occur widely in plant structures, including this cone of queen sago, Cycas circinalis
Early Greek philosophers attempted to explain order in nature, anticipating modern concepts. Plato (c. 427 – c. 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") of which physical objects are never more than imperfect copies. Thus, a flower may be roughly circular, but it is never a perfect mathematical circle.[2] Pythagoras explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.[3] Empedocles to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.[4]
In 1202, Leonardo Fibonacci (c. 1170 – c. 1250) introduced the Fibonacci number sequence to the western world with his book Liber Abaci.[5] Fibonacci gave an (unrealistic) biological example, on the growth in numbers of a theoretical rabbit population.[6]
In 1658, the English physician and philosopher Sir Thomas Browne discussed "how Nature Geometrizeth" in The Garden of Cyrus, citing Pythagorean numerology involving the number 5, and the Platonic form of the quincunx pattern. The discourse's central chapter features examples and observations of the quincunx in botany.[7]
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, is classic. He showed that simple equations could describe all the apparently complex spiral growth patterns of animal horns and mollusc shells.[8]
The Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams.[9]
The German psychologist Adolf Zeising (1810–1876) claimed that the golden ratio was expressed in the arrangement of plant parts, in the skeletons of animals and the branching patterns of their veins and nerves, as well as in the geometry of crystals.[10][11][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 (1865–1931) took the first micrograph of a snowflake in 1885.[14]
D'Arcy Thompson pioneered the study of growth and form in his 1917 book
In 1952, Alan Turing (1912–1954), better known for his work on computing and codebreaking, 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.[15] 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.[16]
In 1968, the 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.[17] 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, crystallising mathematical thought into the concept of the fractal.[18]
Causes[edit]
Composite patterns: aphids and newly born young in arraylike clusters on sycamore leaf, divided into polygons by 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.[19] 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.[20]}
Mathematics seeks to discover and explain abstract patterns or regularities of all kinds.[21][22] 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.[17]
The growth patterns of certain trees resemble these Lindenmayer system fractals.
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. Exact mathematical perfection can only approximate real objects.[23] Visible patterns in nature are governed by physical laws; for example, meanders can be explained using fluid dynamics.
In biology, natural selection can cause the development of patterns in living things for several reasons, including camouflage,[24] sexual selection,[24] and different kinds of signalling, including mimicry[25] and cleaning symbiosis.[26] In plants, the shapes, colours, and patterns of insect-pollinated flowers like the lily have evolved to attract insects such as bees. Radial patterns of colours and stripes, some visible only in ultraviolet light serve as nectar guides that can be seen at a distance.[27]
Types of pattern[edit]
Symmetry[edit]
Further information: Symmetry in biology, Floral symmetry, and Crystal symmetry
Symmetry is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids.[28] Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies.[29]
Among non-living things, snowflakes have striking sixfold symmetry; each flake's structure forms a record of the varying conditions during its crystallization, with nearly the same pattern of growth on each of its six arms.[30] Crystals in general have a variety of symmetries and crystal habits; they can be cubic or octahedral, but true crystals cannot have fivefold symmetry (unlike quasicrystals).[31] Rotational symmetry is found at different scales among non-living things, including the crown-shaped splash pattern formed when a drop falls into a pond,[32] and both the spheroidal shape and rings of a planet like Saturn.[33]
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).[34] 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.[35]
Animals often show mirror or bilateral symmetry, like this tiger.
Echinoderms like this starfish have fivefold symmetry.
Fivefold symmetry can be seen in many flowers and some fruits like this medlar.
Snowflakes have sixfold symmetry.
Fluorite showing cubic crystal habit.
Water splash approximates radial symmetry.
Garnet showing rhombic dodecahedral crystal habit.
Volvox has spherical symmetry.
Sea anemones have rotational symmetry.
Trees, fractals[edit]
Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension.[18][36][37] Infinite iteration is not possible in nature so all 'fractal' patterns are only approximate. For example, the leaves of ferns and umbellifers (Apiaceae) are only self-similar (pinnate) to 2, 3 or 4 levels. Fern-like growth patterns occur in plants and in animals including bryozoa, corals, hydrozoa like the air fern, Sertularia argentea, and in non-living things, notably electrical discharges. Lindenmayer system fractals can model different patterns of tree growth by varying a small number of parameters including branching angle, distance between nodes or branch points (internode length), and number of branches per branch point.[17]
Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic fault lines, mountains, coastlines,[38] animal coloration, snow flakes,[39] crystals,[40] blood vessel branching,[41] actin cytoskeleton,[42] and ocean waves.[43]
Leaf of cow parsley, Anthriscus sylvestris, is 2- or 3-pinnate, not infinite
Fractal spirals: Romanesco broccoli showing self-similar form
Angelica flowerhead, a sphere made of spheres (self-similar)
Trees: Lichtenberg figure: high voltage dielectric breakdown in an acrylic polymer block
Trees: dendritic copper crystals (in microscope)
Spirals[edit]
Further information: phyllotaxis
Spirals are common in plants and in some animals, notably 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.[44] Given a modern understanding of fractals, a growth spiral can be seen as a special case of self-similarity.[45]
Plant spirals can be seen in phyllotaxis, the arrangement of leaves on a stem, and in the arrangement (parastichy[46]) of other parts as in composite flower heads and seed heads like the sunflower or fruit structures like the pineapple[47]:337 and snake fruit, as well as in the pattern of scales in pine cones, 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.[48] 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.[49] In disc phyllotaxis as in the sunflower and daisy, the florets are arranged in Fermat's spiral with Fibonacci numbering, at least when the flowerhead is mature so all the elements are the same size. Fibonacci ratios approximate the golden angle, 137.508°, which governs the curvature of Fermat's spiral.[50]
From the point of view of physics, spirals are lowest-energy configurations[51] which emerge spontaneously through self-organizing processes in dynamic systems.[52] 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.[53] 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.[47]
Fibonacci spiral
Bighorn sheep, Ovis canadensis
Spirals: phyllotaxis of spiral aloe, Aloe polyphylla
Nautilus shell's logarithmic growth spiral
Fermat's spiral: seed head of sunflower, Helianthus annuus
Multiple Fibonacci spirals: red cabbage in cross section
Spiralling shell of Trochoidea liebetruti
Water droplets fly off a wet, spinning ball in equiangular spirals
Chaos, flow, meanders[edit]
In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect"[54]), which requires the mathematical properties of topological mixing and dense periodic orbits.[55]
Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the strange attractors in chaotic systems have a fractal dimension.[56] Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30.[57]
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.[58] 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, which 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 like sand and gravel 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.[59]
Chaos: shell of gastropod mollusc the cloth of gold cone, Conus textile, resembles Rule 30 cellular automaton
Chaos: vortex street of clouds
Meanders: dramatic meander scars and oxbow lakes in the broad flood plain of the Rio Negro, seen from space
Meanders: sinuous path of Rio Cauto, Cuba
Meanders: sinuous snake crawling
Meanders: symmetrical brain coral, Diploria strigosa
Waves, dunes[edit]
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.[60] 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.[61] 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.[62]
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 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.[63] Apart from this nonlinearity, barchans behave rather like solitary waves.[64]
Waves: breaking wave in a ship's wake
Dunes: sand dunes in Taklamakan desert, from space
Dunes: barchan crescent sand dune
Wind ripples with dislocations in Sistan, Afghanistan
Bubbles, foam[edit]
A soap bubble forms a sphere, a surface with minimal area — 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.[9]
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).[65][66] Structures with minimal surfaces can be used as tents. Lord Kelvin identified the problem of the most efficient way to pack cells of equal volume as a foam in 1887; his solution uses just one solid, the bitruncated cubic honeycomb with very slightly curved faces to meet Plateau's laws. No better solution was found until 1993 when Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure; the Beijing National Aquatics Center adapted the structure for their outer wall in the 2008 Summer Olympics.[67]
At the scale of living cells, foam patterns are common; radiolarians, sponge spicules, silicoflagellate exoskeletons and the calcite skeleton of a sea urchin, Cidaris rugosa, all resemble mineral casts of Plateau foam boundaries.[68][69] The skeleton of the Radiolarian, Aulonia hexagona, a beautiful marine form drawn by Ernst Haeckel, looks as if it is a sphere composed wholly of hexagons, but this is mathematically impossible. The Euler characteristic states that for any 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).[70]
Foam of soap bubbles: four edges meet at each vertex, at angles close to 109.5°, as in two C-H bonds in methane.
Radiolaria drawn by Haeckel in his Kunstformen der Natur (1904).
Haeckel's Spumellaria; the skeletons of these Radiolaria have foam-like forms.
Buckminsterfullerene C60: Richard Smalley and colleagues synthesised the fullerene molecule in 1985.
Brochosomes (secretory microparticles produced by leafhoppers) often approximate fullerene geometry.
Circus tent approximates a minimal surface.
Beijing's National Aquatics Center for the 2008 Olympic games has a Weaire–Phelan structure.
Equal spheres (gas bubbles) in a surface foam
Tessellations[edit]
Main article: tessellation
Tessellations are patterns formed by repeating tiles all over a flat surface. There are 17 wallpaper groups of tilings.[71] 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 wax cells in honeycomb built by honey bees are well-known examples. Among animals, 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. Among flowers, the snake's head fritillary, Fritillaria meleagris, have a tessellated chequerboard pattern on their petals. The structures of minerals provide good examples of regularly repeating three-dimensional arrays. Despite the hundreds of thousands of known minerals, there are rather few possible types of arrangement of atoms in a crystal, defined by crystal structure, crystal system, and point group; for example, there are exactly 14 Bravais lattices for the 7 lattice systems in three-dimensional space.[72]
Crystals: cube-shaped crystals of halite (rock salt); cubic crystal system, isometric hexoctahedral crystal symmetry
Arrays: honeycomb is a natural tessellation
Bismuth hopper crystal illustrating the stairstep crystal habit.
Tilings: tessellated flower of snake's head fritillary, Fritillaria meleagris
Tilings: overlapping scales of common roach, Rutilus rutilus
Tilings: overlapping scales of snakefruit or salak, Salacca zalacca
Tessellated pavement: a rare rock formation on the Tasman Peninsula
Cracks[edit]
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.[73] 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.[74]
Old pottery surface, white glaze 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
Drying elastic mud in Sicily with mainly 120° cracks
Cooled basalt at Giant's Causeway. Vertical mainly 120° cracks giving hexagonal columns
Palm trunk with branching vertical cracks (and horizontal leaf scars)
Spots, stripes[edit]
Leopards and ladybirds are spotted; angelfish and zebras are striped.[75] These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce. One function of animal patterns is camouflage;[24] for instance, a leopard that is harder to see catches more prey. Another function is signalling[25] — for instance, a ladybird is less likely to be attacked by 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.[75]
Dirce beauty butterfly, Colobura dirce
Grevy's zebra, Equus grevyi
Royal angelfish, Pygoplites diacanthus
Leopard, Panthera pardus pardus
Array of ladybirds by G.G. Jacobson
Breeding pattern of cuttlefish, Sepia officinalis
Pattern formation[edit]
Main article: Pattern formation
Alan Turing,[15] and later the mathematical biologist James Murray,[76] described a mechanism that spontaneously creates spotted or striped patterns: a reaction-diffusion system.[77] 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.[77]
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).[78] Richard Prum's activation-inhibition models, developed from Turing's work, use six variables to account for the observed range of nine basic within-feather pigmentation patterns, from the simplest, a central pigment patch, via concentric patches, bars, chevrons, eye spot, pair of central spots, rows of paired spots and an array of dots.[79][80]:6 More elaborate models simulate complex feather patterns in the guineafowl 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.[80]:7–8
Patterns can form for other reasons in the vegetated landscape of tiger bush[81] and fir waves.[82] Tiger bush stripes occur on arid slopes where plant growth is limited by rainfall. Each roughly horizontal stripe of vegetation effectively collects the rainwater from the bare zone immediately above it.[81] 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.[82] Natural patterns are sometimes formed by animals, as in the Mima mounds of the Northwestern United States and some other areas, which appear to be created over many years by the burrowing activities of pocket gophers,[83] while the so-called 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.[84]
In permafrost soils with an active upper layer subject to annual freeze and thaw, patterned ground can form, creating circles, nets, ice wedge polygons, steps, and stripes. 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.[85]
The fissured pattern that develops on vertebrate brains are caused by a physical process of constrained expansion dependent on two geometric parameters: relative tangential cortical expansion and relative thickness of 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.[86][87]
Giant pufferfish, Tetraodon mbu
Detail of giant pufferfish skin pattern
Snapshot of simulation of Belousov-Zhabotinsky reaction
Helmeted guineafowl, Numida meleagris, feathers transition from barred to spotted, both in-feather and across the bird
Aerial view of a tiger bush plateau in Niger
Fir waves in White Mountains, New Hampshire
Patterned ground: a melting pingo with surrounding ice wedge polygons near Tuktoyaktuk, Canada
Fairy circles in the Marienflusstal area in Namibia
Human brain (superior view) exhibiting patterns of gyri and sulci
See also[edit]
Emergence
Evolutionary history of plants
Mathematics and art
Widmanstätten pattern
References[edit]
Jump up ^ Stevens 1974, p. 3.
Jump up ^ Balaguer, Mark (7 April 2009) [2004]. "Stanford Encyclopedia of Philosophy". Platonism in Metaphysics. Stanford University. Retrieved 4 May 2012.
Jump up ^ 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 , cc. 350 BC
Jump up ^ 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).
Jump up ^ Singh, Parmanand. Acharya Hemachandra and the (so called) Fibonacci Numbers. Math. Ed. Siwan, 20(1):28–30, 1986. ISSN 0047-6269
Jump up ^ Knott, Ron. "Fibonacci's Rabbits". University of Surrey Faculty of Engineering and Physical Sciences.
Jump up ^ Browne, Thomas (1658). How Nature Geometrizeth. The Garden of Cyrus.
Jump up ^ About D'Arcy. D' Arcy 150. University of Dundee and the University of St Andrews. Retrieved 16 October 2012.
^ Jump up to: a b Stewart 2001, pp. 108–109.
Jump up ^ Padovan, Richard (1999). Proportion. Taylor & Francis. pp. 305–306. ISBN 978-0-419-22780-9.
Jump up ^ Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1): 113–122. doi:10.1007/s00004-001-0008-7.
Jump up ^ Zeising, Adolf (1854). Neue Lehre van den Proportionen des meschlischen Körpers. preface.
Jump up ^ Ball 2009, p. 41.
Jump up ^ Hannavy, John (2007). Encyclopedia of Nineteenth-Century Photography. 1. CRC Press. p. 149. ISBN 978-0-415-97235-2.
^ Jump up to: a b Turing, A. M. (1952). "The Chemical Basis of Morphogenesis". Philosophical Transactions of the Royal Society B. 237 (641): 37–72. Bibcode:1952RSPTB.237...37T. doi:10.1098/rstb.1952.0012.
Jump up ^ Ball 2009, pp. 163, 247–250.
^ Jump up to: a b c Rozenberg, Grzegorz; Salomaa, Arto. The Mathematical Theory of L Systems. Academic Press, New York, 1980. ISBN 0-12-597140-0
^ Jump up to: a b Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan.
Jump up ^ Forbes, Peter. All that useless beauty. The Guardian. Review: Non-fiction. 11 February 2012.
Jump up ^ Stevens 1974, p. 222.
Jump up ^ Steen, L.A.. The Science of Patterns. Science, 240: 611–616, 1998. Summary at ascd.org
Jump up ^ Devlin, Keith. Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996
Jump up ^ Tatarkiewicz, Władysław. Perfection in the Sciences. II. Perfection in Physics and Chemistry, Dialectics and Humanism, vol. VII, no. 2 (spring 1980), p. 139.
^ Jump up to: a b c Darwin, Charles. On the Origin of Species. 1859, chapter 4.
^ Jump up to: a b Wickler, Wolfgang (1968). Mimicry in plants and animals. New York: McGraw-Hill.
Jump up ^ Poulin, R.; Grutter, A.S. (1996) "Cleaning symbiosis: proximate and adaptive explanations". Bioscience 46(7): 512–517. (subscription required)
Jump up ^ Koning, Ross (1994). "Plant Physiology Information Website". Pollination Adaptations. Ross Koning. Retrieved May 2, 2012.
Jump up ^ Stewart 2001, pp. 48–49.
Jump up ^ Stewart 2001, pp. 64–65.
Jump up ^ Stewart 2001, p. 52.
Jump up ^ Stewart 2001, pp. 82–84.
Jump up ^ Stewart 2001, p. 60.
Jump up ^ Stewart 2001, p. 71.
Jump up ^ Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002). "Animal Diversity" (PDF). Chapter 8: Acoelomate Bilateral Animals (Third ed.). McGraw-Hill. p. 139. Retrieved October 25, 2012.
Jump up ^ Sumrall, Colin D.; Wray, Gregory A. (January 2007). "Ontogeny in the fossil record: diversification of body plans and the evolution of "aberrant" symmetry in Paleozoic echinoderms". Paleobiology. 33 (1): 149–163. doi:10.1666/06053.1. JSTOR 4500143.
Jump up ^ Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley.
Jump up ^ Briggs, John (1992). Fractals:The Patterns of Chaos. Thames and Hudson. p. 148.
Jump up ^ Batty, Michael (1985-04-04). "Fractals – Geometry Between Dimensions". New Scientist. 105 (1450): 31.
Jump up ^ Meyer, Yves; Roques, Sylvie (1993). Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications," Toulouse, France – June 1992. Atlantica Séguier Frontières. p. 25. ISBN 9782863321300.
Jump up ^ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000). Pattern formation in biology, vision and dynamics. World Scientific. p. 78. ISBN 9789810237929.
Jump up ^ Hahn, Horst K.; Georg, Manfred; Peitgen, Heinz-Otto (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. Fractals in biology and medicine. Springer. pp. 55–66.
Jump up ^ Sadegh, Sanaz (2017). "Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork". Physical Review X. 7 (1). doi:10.1103/PhysRevX.7.011031. PMC 5500227  . PMID 28690919.
Jump up ^ Addison, Paul S. (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44–46.
Jump up ^ Maor, Eli. e: The Story of a Number. Princeton University Press, 2009. Page 135.
Jump up ^ Ball 2009, pp. 29–32.
Jump up ^ "Spiral Lattices & Parastichy". Smith College. Archived from the original on 26 May 2010. Retrieved 24 September 2013.
^ Jump up to: a b Kappraff, Jay (2004). "Growth in Plants: A Study in Number" (PDF). Forma. 19: 335–354.
Jump up ^ Ball 2009, p. 13.
Jump up ^ Coxeter, H. S. M. (1961). Introduction to geometry. Wiley. p. 169.
Jump up ^ Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0-387-97297-8.
Jump up ^ Levitov LS (15 March 1991). "Energetic Approach to Phyllotaxis" (PDF). Europhys. Lett. 14 (6): 533–9. Bibcode:1991EL.....14..533L. doi:10.1209/0295-5075/14/6/006. (subscription required)
Jump up ^ Douady, S; Couder, Y. (March 1992). "Phyllotaxis as a physical self-organized growth process". Physical Review Letters. 68 (13): 2098–2101. Bibcode:1992PhRvL..68.2098D. doi:10.1103/PhysRevLett.68.2098. PMID 10045303. (subscription required)
Jump up ^ Ball 2009, pp. 163, 249–250.
Jump up ^ Lorenz, Edward N. (March 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.
Jump up ^ Elaydi, Saber N. (1999). Discrete Chaos. Chapman & Hall/CRC. p. 117.
Jump up ^ Ruelle, David. Chance and Chaos. Princeton University Press, 1991.
Jump up ^ Wolfram, Stephen. A New Kind of Science. Wolfram Media, 2002.
Jump up ^ von Kármán, Theodore. Aerodynamics. McGraw-Hill (1963): ISBN 978-0070676022. Dover (1994): ISBN 978-0486434858.
Jump up ^ 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. Archived from the original (PDF) on 2011-09-29. .
Jump up ^ French, A.P. Vibrations and Waves. Nelson Thornes, 1971.
Jump up ^ Tolman, H.L. (2008), "Practical wind wave modeling", in Mahmood, M.F., CBMS Conference Proceedings on Water Waves: Theory and Experiment (PDF), Howard University, USA, 13–18 May 2008: World Scientific Publ.
Jump up ^ "Types of Dunes". USGS. 29 October 1997. Retrieved May 2, 2012.
Jump up ^ Strahler, A. & Archibold, O.W. Physical Geography: Science and Systems of the Human Environment. John Wiley, 4th edition 2008. Page 442.
Jump up ^ Schwämmle, V.; Herrman, H.J. (2003). "Solitary wave behaviour of sand dunes". Nature. 426 (Dec. 11): 619–620 Abstract. Bibcode:2003Natur.426..619S. doi:10.1038/426619a. PMID 14668849. (subscription required)
Jump up ^ Ball 2009, p. 68.
Jump up ^ Frederick J. Almgren, Jr. and Jean E. Taylor, The geometry of soap films and soap bubbles, Scientific American, vol. 235, pp. 82–93, July 1976.
Jump up ^ Ball 2009, pp. 73–76.
Jump up ^ Ball 2009, pp. 96–101.
Jump up ^ Brodie, Christina (February 2005). "Geometry and Pattern in Nature 3: The holes in radiolarian and diatom tests". Microscopy-UK. Retrieved May 28, 2012.
Jump up ^ Ball 2009, pp. 51–54.
Jump up ^ Armstrong, M.A. (1988). Groups and Symmetry. New York: Springer-Verlag.
Jump up ^ Hook, J.R.; Hall, H.E. Solid State Physics (2nd Edition). Manchester Physics Series, John Wiley & Sons, 2010. ISBN 978-0-471-92804-1
Jump up ^ Stevens 1974, p. 207.
Jump up ^ Stevens 1974, p. 208.
^ Jump up to: a b Ball 2009, pp. 156–158.
Jump up ^ Murray, James D. (9 March 2013). Mathematical Biology. Springer Science & Business Media. pp. 436–450. ISBN 978-3-662-08539-4.
^ Jump up to: a b Ball 2009, pp. 159–167.
Jump up ^ Ball 2009, pp. 168–180.
Jump up ^ Rothenberg 2011, pp. 93–95.
^ Jump up to: a b Prum, Richard O.; Williamson, Scott (2002). "Reaction–diffusion models of within-feather pigmentation patterning" (PDF). Proceedings of the Royal Society of London B. 269 (1493): 781–792. doi:10.1098/rspb.2001.1896. PMC 1690965  . PMID 11958709.
^ Jump up to: a b Tongway, D.J.; Valentin, C. & Seghieri, J. (2001). Banded vegetation patterning in arid and semiarid environments. New York: Springer-Verlag.
^ Jump up to: a b D'Avanzo, C. (22 February 2004). "Fir Waves: Regeneration in New England Conifer Forests". TIEE. Retrieved 26 May 2012.
Jump up ^ Morelle, Rebecca. "'Digital gophers' solve Mima mound mystery". BBC News. Retrieved 9 December 2013.
Jump up ^ Sample, Ian. "The secret of Namibia's 'fairy circles' may be explained at last". The Guardian. Retrieved 18 January 2017.
Jump up ^ "Permafrost: Patterned Ground". US Army Corps of Engineers. Retrieved 17 February 2015.
Jump up ^ Ghose, Tia. "Human Brain's Bizarre Folding Pattern Re-Created in a Vat". Retrieved 5 April 2018.
Jump up ^ Tallinen, Tuoma; Chung, Jun Young; Biggins, John S.; Mahadevan, L. (2014). "Gyrification from constrained cortical expansion". Proceedings of the National Academy of Sciences of the United States of America. 111 (35): 12667–12672.
Bibliography[edit]
Pioneering authors[edit]
Latin Wikisource has original text related to this article:
Liber abbaci
Fibonacci, Leonardo. Liber Abaci, 1202.
----- translated by Sigler, Laurence E. Fibonacci's Liber Abaci. Springer, 2002.
Haeckel, Ernst. Kunstformen der Natur (Art Forms in Nature), 1899–1904.
Thompson, D'Arcy Wentworth. On Growth and Form. Cambridge, 1917.
General books[edit]
Adam, John A. Mathematics in Nature: Modeling Patterns in the Natural World. Princeton University Press, 2006.
Ball, Philip (2009). Nature's Patterns: a tapestry in three parts. 1:Shapes. Oxford University Press.
Ball, Philip (2009). Nature's Patterns: a tapestry in three parts. 2:Flow. Oxford University Press.
Ball, Philip (2009). Nature's Patterns: a tapestry in three parts. 3. Branches. Oxford University Press.
Ball, Philip. Patterns in Nature. Chicago, 2016.
Murphy, Pat and Neill, William. By Nature's Design. Chronicle Books, 1993.
Rothenberg, David (2011). Survival of the Beautiful: Art, Science and Evolution. Bloomsbury Press.
Stevens, Peter S. (1974). Patterns in Nature. Little, Brown & Co.
Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson.
Patterns from nature (as art)[edit]
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.
Porter, Eliot, and Gleick, James. Nature's Chaos. Viking Penguin, 1990.
External links[edit]
Fibonacci Numbers and the Golden Section
Phyllotaxis: an Interactive Site for the Mathematical Study of Plant Pattern Formation
hide
v
t
e
Patterns in nature
Patterns
Crack
Dune
Foam
Meander
Phyllotaxis
Soap bubble
Symmetry
in crystals
Quasicrystals
in flowers
in biology
Tessellation
Vortex street
Wave
Widmanstätten pattern
Causes
Pattern formation
Biology
Natural selection
Camouflage
Mimicry
Sexual selection
Mathematics
Chaos theory
Fractal
Logarithmic spiral
Physics
Crystal
Fluid dynamics
Plateau's laws
Self-organization
People
Plato
Pythagoras
Empedocles
Fibonacci
Liber Abaci
Adolf Zeising
Ernst Haeckel
Joseph Plateau
Wilson Bentley
D'Arcy Wentworth Thompson
On Growth and Form
Alan Turing
The Chemical Basis of Morphogenesis
Aristid Lindenmayer
Benoît Mandelbrot
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
Related
Pattern recognition
Emergence
Mathematics and art
Retrieved from "https://en.wikipedia.org/w/index.php?title=Patterns_in_nature&oldid=856929409"
Categories:
Patterns
Nature
Applied mathematics
Recreational mathematics
Pattern formation
Hidden categories:
Pages containing links to subscription-only content
Good articles
Articles containing Ancient Greek-language text
Navigation menu
Personal tools
Not logged in
Talk
Contributions
Create account
Log in
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
More
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikipedia store
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Wikidata item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
العربية
Eesti
Español
Français
Հայերեն
עברית
Norsk nynorsk
Русский
ไทย
Українська
3 more
Edit links
This page was last edited on 28 August 2018, at 13:01 (UTC).
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Developers
Cookie statement
Mobile view
Enable previews

values: mw-page-base noprint

values: mw-head-base noprint

values: This is a good article. Follow the link for more information.

attrs: class id

values: firstHeading firstHeading en

values: siteSub noprint

attrs: class href

values: mw-jump-link #mw-head

values: mw-jump-link #p-search

value: p

attrs: alt src width height class srcset data-file-width data-file-height

attrs: href class title

values:
/wiki/Namib_Desert
mw-redirect
Namib Desert

values:
/wiki/Modelled_mathematically
mw-redirect
Modelled mathematically

values:
/wiki/Greek_philosophers
mw-redirect
Greek philosophers

values:
/wiki/D%27Arcy_Thompson
mw-redirect
D'Arcy Thompson

values:
/wiki/Turing_patterns
mw-redirect
Turing patterns

values:
/wiki/Beno%C3%AEt_Mandelbrot
mw-redirect
Benoît Mandelbrot

values:
/wiki/Computer_models
mw-redirect
Computer models

values: checkbox button toctogglecheckbox toctogglecheckbox display:none

values:
/wiki/Universals_(metaphysics)
mw-redirect
Universals (metaphysics)

values:
/wiki/Theory_of_Forms
mw-redirect
Theory of Forms

values:
/wiki/Sir_Thomas_Browne
mw-redirect
Sir Thomas Browne

values:
/wiki/Pythagorean_numerology
mw-redirect
Pythagorean numerology

values:
/wiki/Platonic_form
mw-redirect
Platonic form

values:
/wiki/Fibonacci_sequence
mw-redirect
Fibonacci sequence

values:
/wiki/Animal_horn
mw-redirect
Animal horn

values:
/wiki/Gottfried_Leibniz
mw-redirect
Gottfried Leibniz

values:
/wiki/Vein_(botany)
mw-redirect
Vein (botany)

values:
/wiki/Insect_pollination
mw-redirect
Insect pollination

values:
/wiki/European_honey_bee
mw-redirect
European honey bee

values:
/wiki/Nectar_guides
mw-redirect
Nectar guides

values:
/wiki/Crystal_symmetry
mw-redirect
Crystal symmetry

values:
/wiki/Dihedral_symmetry
mw-redirect
Dihedral symmetry

values:
/wiki/Bilateral_symmetry
mw-redirect
Bilateral symmetry

values:
/wiki/Pentaradial_symmetry
mw-redirect
Pentaradial symmetry

values:
/wiki/Radial_symmetry
mw-redirect
Radial symmetry

values:
/wiki/Electrical_discharge
mw-redirect
Electrical discharge

values:
/wiki/Internode_(botany)
mw-redirect
Internode (botany)

values:
/wiki/Actin_cytoskeleton
mw-redirect
Actin cytoskeleton

values:
/wiki/Dynamic_system
mw-redirect
Dynamic system

values:
/wiki/Equiangular_spiral
mw-redirect
Equiangular spiral

values:
/wiki/Topological_mixing
mw-redirect
Topological mixing

values:
/wiki/Periodic_orbits
mw-redirect
Periodic orbits

values:
/wiki/Strange_attractors
mw-redirect
Strange attractors

values:
/wiki/Cellular_automata
mw-redirect
Cellular automata

values:
/wiki/Vortex_street
mw-redirect
Vortex street

values:
/wiki/Helical_flow
mw-redirect
Helical flow

values:
/wiki/Positive_feedback_loop
mw-redirect
Positive feedback loop

values:
/wiki/Flood_plain
mw-redirect
Flood plain

values:
/wiki/Lord_Kelvin
mw-redirect
Lord Kelvin

values:
/wiki/Spicule_(sponge)
mw-redirect
Spicule (sponge)

values:
/wiki/Convex_polyhedron
mw-redirect
Convex polyhedron

values:
/wiki/Bravais_lattices
mw-redirect
Bravais lattices

values:
/wiki/Rutilus_rutilus
mw-redirect
Rutilus rutilus

values:
/wiki/Grevy%27s_zebra
mw-redirect
Grevy's zebra

values:
/wiki/Georgij_Georgiewitsch_Jacobson
mw-redirect
Georgij Georgiewitsch Jacobson

values:
/wiki/Numida_meleagris
mw-redirect
Numida meleagris

values:
/wiki/Pocket_gopher
mw-redirect
Pocket gopher

values:
/wiki/Fairy_circle
mw-redirect
Fairy circle

values:
/wiki/Thermal_contraction
mw-redirect
Thermal contraction

values:
/wiki/Cerebellar_cortex
mw-redirect
Cerebellar cortex

values:
/wiki/Giant_pufferfish
mw-redirect
Giant pufferfish

values:
/wiki/Belousov-Zhabotinsky_reaction
mw-redirect
Belousov-Zhabotinsky reaction

values:
nofollow
external text
http://www.darcythompson.org/about.html

values: Philosophical Transactions of the Royal Society B

values:
nofollow
external text
//doi.org/10.1098/rstb.1952.0012

values:
/wiki/Rozenberg,_Grzegorz
mw-redirect
Rozenberg, Grzegorz

values:
nofollow
external text
https://www.jstor.org/stable/1312929

values:
nofollow
external text
//doi.org/10.1666/06053.1

values:
nofollow
external text
//www.jstor.org/stable/4500143

values:
nofollow
external text
//doi.org/10.1103/PhysRevX.7.011031

attrs: alt src title width height srcset data-file-width data-file-height

values:
/wiki/PubMed_Identifier
mw-redirect
PubMed Identifier

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/28690919

values:
/wiki/Przemyslaw_Prusinkiewicz
mw-redirect
Przemyslaw Prusinkiewicz

values:
nofollow
external text
//doi.org/10.1103/PhysRevLett.68.2098

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/10045303

values:
nofollow
http://pubs.usgs.gov/gip/deserts/dunes/
external text

values:
nofollow
external text
https://dx.doi.org/10.1038/426619a

values:
nofollow
external text
//doi.org/10.1038/426619a

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/14668849

values:
/wiki/Frederick_J._Almgren,_Jr.
mw-redirect
Frederick J. Almgren, Jr.

values:
/wiki/Jean_E._Taylor
mw-redirect
Jean E. Taylor

values:
nofollow
external text
//doi.org/10.1098/rspb.2001.1896

values:
nofollow
external text
//www.ncbi.nlm.nih.gov/pubmed/11958709

values: -moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;

values:
/wiki/Bloomsbury_Press
mw-redirect
Bloomsbury Press

values:
/wiki/Little,_Brown_%26_Co
mw-redirect
Little, Brown & Co

values: Edit section: Patterns from nature (as art)

values:
/wiki/Porter,_Eliot
mw-redirect
Porter, Eliot

values:
/wiki/Viking_Penguin
mw-redirect
Viking Penguin

values: Template talk:Patterns in nature (page does not exist)

values:
navbox-image
4
width:1px;padding:0px 0px 0px 2px

values: nowraplinks hlist collapsible autocollapse navbox-inner mw-collapsible mw-made-collapsible

values: Category:Pages containing links to subscription-only content

values: Category:Articles containing Ancient Greek-language text

values: Discussion about edits from this IP address [ctrl-alt-n]

values: A list of edits made from this IP address [ctrl-alt-y]

values: You are encouraged to create an account and log in; however, it is not mandatory

values:
/wiki/Patterns_in_nature
View the content page [ctrl-alt-c]
c

values: About the project, what you can do, where to find things

values: A list of recent changes in the wiki [ctrl-alt-r]

values: portal navigation p-interaction p-interaction-label

values: List of all English Wikipedia pages containing links to this page [ctrl-alt-j]

values: Recent changes in pages linked from this page [ctrl-alt-k]

values:
/wiki/Wikipedia:File_Upload_Wizard
Upload files [ctrl-alt-u]
u

values: A list of all special pages [ctrl-alt-q]

values: Permanent link to this revision of the page

values: Link to connected data repository item [ctrl-alt-g]

values: Printable version of this page [ctrl-alt-p]

values: أنماط في الطبيعة – Arabic

values: Բնության օրինաչափություններ – Armenian

values: תבניות בטבע – Hebrew

values:
interlanguage-link interwiki-nn
display: none;

values: Закономерности в природе – Russian

values: ลวดลายในธรรมชาติ – Thai

values:
interlanguage-link interwiki-th
display: none;

values: Патерни у природі – Ukrainian

values:
interlanguage-link interwiki-uk
display: none;

values: All languages (initial selection from common choices by you and others)

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Patterns_in_nature rootpage-Patterns_in_nature skin-vector action-view