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  Turing's  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:        

           

       

                                       Turing's  mathematical  model  of  chemical  morphogenesis  helps  us  understand  why  tigers  and  zebras  have  stripes.  Turing's  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>