Phyllotaxis

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Phyllotaxis

• A mathematical theory?
• Sjors van der Stelt

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Fibonacci sequence

• F_11F_21F_n2 F_n1 F_n.
• Gives 1,1,2,3,5,8,13,21,34,55,89,144,233, .
• F_n1/F_n converges to f 0.61803 which is a
  root of x²x-1
  0.This is the golden section. The corresponding
  angle
• f360º 137.5 º is called the golden angle.
• Other elementary property gcd(f_n1,f_n)1

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Phyllotactic Patterns in nature
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Wrong picture taken from De Andere Linkerkant
Links en rechts in de evolutie from Tijs
Goldschmidt
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Nobody understands?

• From Tijs Goldschmidt
• Door Goulds essay over Thompson nam ik ook
  voor het eerst kennis van Fibonacci-reeksen. Een
  Fibonacci-reeks vorm je door steeds de twee
  vorige getallen in de reeks bij elkaar op te
  tellen en daarmee het volgende getal te vormen.
  Dus 1, 1, 2, 3, 5, 8, 13, 21. () Zon
  dennenkegel bijvoorbeeld kan, als je de
  afzonderlijke schubben volgt, zijn opgebouwd uit
  13 links- en 21-rechtsdraaiende spiralen. Dus
  niet 16 en 24, of 9 en 19, maar uitgerekend 13 en
  21, allebei getallen uit de Fibonacci-reeks.
  Waarom volgen zoveel structuren deze curve die
  wiskundigen een logaritmische spiraal noemen?
  Thompsons antwoord op de vraag was destijds
  geweest dat deze abstracte vormen de best
  mogelijke oplossingen zijn voor algemene
  problemen.

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(Serious) literature on phyllotaxis

• Kepler and Leonardo were among the first to
  recognize the phenomenon
• Extensively studied during nineteenth en
  twentieth century articles were mostly published
  in German or French during the nineteenth
  century, later English. However, English and
  French people rarely speak German!!(to be
  continued)

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Preliminaries

• The seeds or petals that form the phyllotactic
  patterns are called primordia.
• There are n spirals, called parastichies,
  spiralling to the one direction (right, par
  exemple) and m parastichies going to the left.
  (Of course, there are others, but these are the
  best discernable. They are also called contact
  parastichies.)
• If we number the primordia from youngest to
  oldest, then two neighbours on a n-parastichy
  differ n.
• The angle between two primordium and the next
  oldest primordium is called the divergence
  angle
• Wrong model fixed divergence angle radial
  growth of vk
• Assume divergence angle d is a rational number
  (as a fraction of the number of revolutions), for
  example d 0.125
• Bravais Bravais (1830s) proved that if a
  phyllotactic structure has d f then the
  parastichy numbers are equal to Fibonacci numbers
• Tait (1872), Coxeter (70s) (pictures taken from
  Michael Naylor 02)

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Numbering of primordia
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We then get
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Other examples
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What if d f?
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And what about d v2?
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What if we choose d p?
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So d p does not give an equal distribution of
the petals on the disk, but f and v2 do. Why?

• The continued fraction expansion of p converges
  relatively rapidly
• Now f and v2 have continued fraction expansions
  that converge slowly (according to a criterion of
  Klein)

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Hofmeisters rules (1870s)

• (Expended version Snow Snow (1950s), Couder
  Douady (1996))
• The apex has an axis of symmetry.
• Primordia form at edge of the apex and, due to
  the shoots growth, they move away radially from
  the center with radial velocity only depending on
  their distance to the edge of the apex.
• New primordia are formed periodically the period
  is called the plastochrone.
• The incipient primordium forms in the largest
  available space left by the previous ones.

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Van Iterson (1907)

• Mathematische und mikroskopische-anatomische
  Studien uber Blattstellungen
• Es besteht natürlich die Möglichkeit, daß die
  Schraubenlinie durch die Punkte a_0 und b_0 alle
  Punkte des Systems umfaßt aber im allgemeinen
  wird dieses nicht eintreffen, und es wird
  anzunehmen sein, daß Punkte außerhalb derselben
  liegen. Wählen wir den Punkt g_0, der von allen
  diesen Punkten am dichtsten an der betrachteten
  Schraubenlinie liegt, und schrauben wir ... (p9)
  (Etcetera etc.!)
• Van Itersons (nor Schoutes) theory is not
  mentioned as often as it should be in the
  abundant literature on phyllotaxis. This might be
  due to the fact that he wrote his thesis in
  German.

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General principle

• There are n spirals, called parastichies,
  spiralling to the one direction (right, par
  exemple) and m parastichies going to the left.
  (Of course, there are others, but these are the
  best discernable. They are also called contact
  parastichies.)
• If we number the primordia from youngest to
  oldest, then two neighbours on a n-parastichy
  differ n.
• If a primordium k on an n-parastichy gets farther
  from the apex, it looses contact with the its
  younger neighbour k-n a bifurcation happens

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Numbering of primordia
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Some calculations from Van Iterson
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Some calculations from Van Iterson
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Van Iterson diagram
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Douady Couder (1996) experimental results

• Douady Couder settled out an experiment in
  which magnetized oil droplets were periodically
  dropped on a disk. The experiments showed
  phyllotactic patterns to a high degree.
• Numerical results published in the same sequence
  of articles showed the same.
• A dynamical model proposed by the authors gave
  rise to a bifurcation diagram which is exactly
  the Van Iterson diagram

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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Dynamical systems approach
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Further theories

• Turing (52) morphogenesis
• Cellular automata
• Lindenmayer systems

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Main References

• Atela P, Gole C and Hotton, S. A Dynamical System
  for Plant Pattern Formation. Nonlinear Science
  2002
• Douady S and Couder Y, Phyllotaxis as a Self
  Organising Process Part I,II and III. Journal of
  Theoretical Biology 1996
• Van Iterson, G., Mathematische und
  mikroskopische-anatomische Studien uber
  Blattstellungen. Jena Verlag 1907