Alan Turing Fibonacci Numbers in Nature Melissa Davis: Alan Turing was a mathematical genius. He speculated that there was a relationship between math and nature by the presence of Fibonacci numbers that naturally occur in plants. Fibonacci numbers are a sequence of numbers, where you can add one of the numbers with the number to the right of it, to get the next number. For example, the first few numbers of the sequence begin as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. To get the number after 21, simply add 13 to 21, which gives you 34. These numbers may correlate to the number of petals, leaves, or spirals of seeds a plant has. This is also the reason why four-‐leaf clovers are so rare, since four is not a number k appears in the Fibonacci sequence. Turing specifically looked at sunflowers to study this phenomenon. Turing examined how the number of spirals in the seed patterns of sunflowers typically resulted in a Fibonacci sequence. This finding was significant, as it provided much information for phyllotaxis, the study of the way plants grow. “The appearance of patterns in the phyllotaxis -‐ the arrangement of leaves, stems, seeds or similar -‐ has been studied by many well-‐known scientists, including Leonardo Da Vinci.” – BBC News
Here is a demonstration of one way to count the spirals of seeds. The total number of rows is 34, which is a Fibonacci number. Because of Alan Turing’s abbreviated life due to his mistreatment in society, Turing was never able to confirm his findings. However, the Manchester Science Festival, the Museum of Science and Industry, and the University of Manchester are asking for help from the public to confirm Turing’s work on Fibonacci numbers in sunflowers. The project entails people planting and growing their own sunflowers, and then counting the rows of seeds as Turing did. This project also aims to honor Turing during the one-‐hundred-‐year anniversary of his death. In addition to sunflowers, the Fibonacci sequence is found in the number of petals in many flowers. For example, buttercups, wild roses, and larkspurs have five petals; delphiniums and coreopsis have eight petals; ragworts and marigolds have thirteen petals; and daisies can have eighty-‐nine petals.
References: Couder, Yves. "Sunflower." Photo. Flower Patterns and Fibonacci Numbers. 2002. 22 May 2012. <http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html>. "The Fibonacci Series." ThinkQuest. Oracle Foundation. Web. 22 May 2012. <http://library.thinkquest.org/27890/applications5.html>. "Greater Manchester Sunflowers to Test Alan Turing Theory." BBC News. BBC, 22 Mar. 2012. Web. 06 June 2012. <http://www.bbc.co.uk/news/uk-‐england-‐manchester-‐17469241>. GrrlScientist. "Sunflowers and Fibonacci." The Guardian. Guardian News and Media, 29 Mar. 0016. Web. 22 May 2012. <http://www.guardian.co.uk/science/grrlscientist/2012/apr/16/1>. "How To Count the Spirals." Photo. Museum of Mathematics. 22 May 2012. <http://momath.org/home/fibonacci-‐numbers-‐of-‐sunflower-‐seed-‐spirals>. McKay, Dennis. "Larkspur." Photo. Drug Discovery. 2009. 22 May 2012. <http://digitalunion.osu.edu/r2/summer09/jaeger/MLA.html>. “Sunflowers and Fibonacci – Numberphile.” Video. (2012). Retrieved May 22, 2012 from
http://www.youtube.com/watchfeature=player_embedded&v=DRjFV_DETKQ. Wainwright, Martin. "Grow a Sunflower to Solve Unfinished Alan Turing Experiment." The Guardian. Guardian News and Media, 24 Nov. 0048. Web. 22 May 2012. <http://www.guardian.co.uk/uk/the-‐northerner/2012/mar/26/alan-‐turing-‐sunf....> Jing (Sophie) Xia: However, flowers are not the only organisms in which Fibonacci numbers are present; Fibonacci numbers are also found in pine cones and plant leaves. Pine cones display the Fibonacci Spirals clearly. The best way to examine these patterns is to observe pine cones from the base where the stalk connects it to the tree. For instance, one set of spirals goes in one uniform direction whereas another set of spirals goes in the opposite direction (see images below). For example, in one direction, there are 8 whirls whereas in the other direction, there are 13 whirls. It is not coincidence that both 8 and 13 are Fibonacci numbers. Pine cones contain evidence of Fibonacci spirals since their patterns are arranged in two different directions of spirals.
In addition, many plants show Fibonacci numbers in the arrangements of the leaves around their stems. When looking down on a plant, one can notice that its leaves are arranged so that the leaves higher up on the stem do not hide leaves below. This ensures that no matter where the leaves are located on a stem, they are able to receive sunlight. Fibonacci numbers are evident in two ways in terms of leaves per turn. First, they occur when counting the number of times they go around the stem. Secondly, it occurs when counting leaves until finding a leaf directly above the leaf in which one started counting. If one counts in the opposite direction, there is a different number of turns with the same number of leaves. The number of turns in each direction and the number of leaves met are three consecutive Fibonacci numbers. For example, one must rotate three turns clockwise to meet a leaf that is directly above the first leaf counted. On the way, one passes by five leaves. But when one counts counter-‐clockwise, they only turn two times. Because 2, 3, and 5 are consecutive Fibonacci numbers, this example demonstrates the existence of Fibonacci numbers in plant leaves. References: "Evolution." How Stuff Works. N.p., n. d. Web. 20 May. 2012. <http://science.howstuffworks.com/environmental/life/evolution>. "Fibonacci Numbers and Nature." Rabbits, Cows and Bees Family Trees . N.p., n. d. Web. 20 May. 2012. <http://www.maths.surrey.ac.uk/hosted-‐sites/R.Knott/Fibonacci/fibnat.html>. "Fibonacci Numbers and the Golden Section." N.p., n. d. Web. 20 May. 2012. <http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm>. "Fibonacci numbers and Golden ratio." Natural occurrence of Fibonacci numbers. N.p., n. d. Web. 20 May. 2012. <http://gwydir.demon.co.uk/jo/numbers/interest/golden.htm>.
Parveen, Nikhat. "Fibonacci in Nature." N.p., n. d. Web. 20 May. 2012. <http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm>. Shiwei Huang Alan Turing’s interest in Fibonacci series was inspired by zoologist D’arcy Wentworth Thompson’s book, On Growth and Form. Thompson wanted to explain how physical and mathematical laws could explain the forms and patterns of living things. After examining the patterns of fir cones and sunflowers, Thompson observed that the scales of a fir cone and the florets of a sunflower are grouped in the numbers in Fibonacci series. However, Thompson claimed that the appearance of Fibonacci series in these plants were purely for mathematical reasons, and the purpose of introducing of this series into plants throughout the course of natural selection was not worth studying. Alan Turing was the first scientist to study the mechanisms behind the development of pattern in living organisms using computer simulation. When Manchester Electronic Computer, also called Ferranti Mark I, was installed in Manchester University, Turing wrote to his colleague, “I am hoping as one of the first jobs to do something about ‘chemical embryology.’ In particular I think one can account for the appearance of Fibonacci numbers in connection with fir cones.” Turing formulated his reaction-‐diffusion model from the observation of fir cones and sunflowers. He believed that diffusing chemicals reacted with each other and caused the development of forms of living organisms. He postulated that the reaction-‐diffusion model could be applied to gastrulation of an embryo, which is the rearranging of the cells in an embryo, and the formation of leaf pattern. Modern computer has simulated Turing’s reaction-‐diffusion mechanism, and it has successfully produced leopard-‐like, cheetah-‐like, and giraffe-‐like stripes. In his paper “Chemical basis of morphogenesis,” he called these interacting chemicals “morphogens.” Unfortunately, Turing’s work on morphogenesis was considered ahead of the time, and he died and left a large number of research materials and notes that could not be understood today. Turing’s reaction-‐diffusion system (the simplified version): ∂c/∂t=f(c)+D∇2c f(c) represents the local chemical reaction that different chemicals are reacted and formed. D is the diffusion constant, which describes the flow of a chemical due to its concentration gradient and its diffusion. Simply put, the equation states that the distribution of a chemical is determined by the chemical reaction that generates this chemical and the diffusion of this chemical. Under certain conditions, two or more chemicals will diffuse and react with each other in the embryo, and they will reach a stable pattern of concentration. For instance, if there was a ring of cells, reaction-‐diffusion model could give us a pattern of chemical gradient surrounding the ring of cells. The same concentrations would occur at places with the same distances to each other, and Turing called this chemical wave if it was stationary. Conversely, if the gradient was changing, it would be called traveling waves. Turing pointed out the structure of the embryo could break this pattern and cause asymmetrical chemical waves. Turing believed that genes catalyzed the production of morphogens and might influence the rate of the reaction to determine the pattern in animals.
Program sheet written by Alan Turing during his study of fir cone patterning ’ Computer output. Turing wrote “How did this happen?” on the sheet. Turing’s numbering on the sunflower References: Charvoin, J. and Sadoc, J-‐F. (2011) “A Phyllotactic Approach to The Structure of Collagen Fibrils.” <http://arxiv.org/pdf/1102.2359v2.pdf>
Copeland, B.J.(2004) “The Essential Turing, Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma.” Oxford: Clarendon Press. Engelhardt, R. (1994) “Modeling Pattern Formation in Reaction-‐Diffusion System.” <http://www.robinengelhardt.info/speciale/main.pdf> Maini, P. K. (2007) “The impact of Turings work on pattern formation in biology.” <people.maths.ox.ac.uk/maini/PKM%20publications/172.pdf> Swinton, J. (2003) “Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis.” <user29459.vs.easily.co.uk/wp-‐content/uploads/2011/05/swinton.pdf> Thompson, D. W. (1966) “On Growth and Form.” Cambridge: University Press. Turing, A. M. (1952) “The Chemical Basis of Morphogensis.” <http://links.jstor.org/sici?sici=0080-‐4622%2819520814%29237%3A641%3C37%3ATCBOM%3E2.0.CO%3B2-‐I Jen-‐Ling Nieh: Morphology, at the most basic sense, consists of two aspects – shape and pattern. The changes of morphology that occur during biological development of an organism are called “morphogenesis.” Turing proposed that both shape and pattern seem to be set up in embryos by the same mechanism, a pre-‐pattern of chemical changes that waits for the appropriate stage of development, and then triggers either pigments, to create pattern, or cellular changes, to create shape. He showed that this kind of system may have a homogeneous stationary state which is unstable against perturbations, such that any random deviation from the stationary state leads through diffusion to a symmetry break. This process is called diffusion-‐driven instability. Since complex spatial patterns are commonly found in nature, for example, in animal skins and also in some polymer systems, it is quite natural to think that such pattern formations could be caused by some general physicochemical process.
Many animals develop their coat patterns in stages. Typically, a secondary pattern will emerge as the animal transitions to adulthood. The following examples all use multiple stages: Turings mathematical model of chemical morphogenesis helps us understand why tigers and zebras have stripes. Turings Reaction-‐Diffusion model from 1952 consists on a set of equations that iteratively simulate the distribution of a chemical agent (activator) modulated by the presence of another agent called inhibitor. In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniform solution of reacting chemicals to develop stable spatially periodic concentration patterns. It is believed that such interactions take place in nature to form patterns that can be found in mammals and fish, and the first model, generating spots.
According to the Reaction-‐Diffusion Model, the diffusion of an activator and inhibitor through an evolving cellular system over a period of time, the concentration gradients dictating cell differentiation, i.e. a zebra skin cell can be black or white according to the concentration of a white-‐cell activator at the point when it forms. Biologists would call these activators morphogens, as these are the proteins that regulate gene expression.
Currently, pigmentation patterns in animal skins, feathers of birds, and shells of snails are the only examples in which we can detect the dynamic nature of Turing waves as a time course of the pattern change. Especially, the two-‐dimensional (2-‐D) skin pattern of fish is quite convenient to study because their waves are sometimes alive even when the fish has grown up into an adult (Shigeru Kondo). For example, when a striped angelfish (Pomacanthus imperator) grows, the branching points of the stripes slide horizontally as the zip opens and add a number of stripes; eventually the spacing between the stripes remains stable. In the case of the spotted catfish (Plecostoms), both division of the spots and insertion of the new spots occur to retain the density and size of the spots. Both stripes and the spots are the most typical 2-‐D patterns generated by the RD mechanism, and the time course of the pattern change possesses the characteristics of the dynamics of RD waves, strongly suggesting that the RD mechanism underlies the process of pigment pattern formation of fish. References: www.scholarpedia.org/w/images/8/8d/TROPH.jpg">http://www.scholarpedia.org/w/images/8/8d/TROPH.jpg http://cgjennings.ca/toybox/turingmorph/texture1.png http://cgjennings.ca/toybox/turingmorph/texture2.png http://cgjennings.ca/toybox/turingmorph/texture3.jpg www.urbagram.net/images/turing.jpg">http://www.urbagram.net/images/turing.jpg www.urbagram.net/v1/revision/Morphogenesis?rev=1">http://www.urbagram.net/v1/revision/Morphogenesis?rev=1 http://27.media.tumblr.com/tumblr_lzeh1qUhRe1r3lyy3o1_500.jpg
Alexandra Pourzia: Alan Turing proposed, based purely on logical reasoning, that pattern formation in nature involved an ‘activating’ substance and an ‘inhibiting’ substance. The repetition of activator and inhibitor could create patterns such as stripes.1 Previously, developmental biologists were puzzled by pattern formation because they could not explain it using the linear models that were the extent of their knowledge at the time. Turing proposed a nonlinear model by introducing diffusion as the generator of instability in the model, instead of being a byproduct of the model. 2 The implications of Turing’s mechanism were astounding: he predicted the mode of action of the Hox genes in Drosophila, which result in the patterning of the embryo’s body segments. 3 Hox gene patterning by body segment in Drosophila The Hox genes induce patterning by activating transcription of their unique set of genes while repressing others not related to their segment. They in turn are regulated by patterning genes (gap, pair-‐rule, or segment polarity genes), which follow Turing’s proposed model very closely. These patterning genes are induced by high or low concentrations of maternal proteins in the embryo, which was formed from the maternal egg and paternal sperm. For example, high concentrations of maternal protein induce the expression of Bicoid and Hunchback, while inhibiting Giant and Kruppel. The concentration of these “morphogens”, as Turing first called them, lead to the formation of a pattern – segment two of the fly embryo.3 Pair-‐rule genes: expressed between certain segments
References: 1. Hughes, Virginia. “Alan Turing’s 60-‐Year-‐Old Prediction About Patterns in Nature Proved True.“ Smithsonian.com. The Smithsonian Institution, 21 Feb 2012. Web. 20 May 2012. <http://blogs.smithsonianmag.com/science/2012/02/alan-‐turing-‐ predicted-‐natures-‐stripes-‐and-‐patterns/> 2. Reinitz, John. “Pattern formation.” Nature. Feb 2012. 3. “Hox gene.” Wikipedia. Wikimedia Foundation, Inc, n.d. 20 May 2012. <http://en.wikipedia.org/wiki/Hox_gene>