MATH0011
Numbers and Patterns in Nature and Life
„
Patterns of arrangements of seeds on a
sunflower head, a pineapple, a pine cone, etc:
Lecture 9
Phyllotaxis –
Fibonacci patterns in plants
http://147.8.101.93/MATH0011/
sunflower head
pinecone
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Phyllotaxis (phyllo = leaf, taxis = arrangement):
study of geometrical and numerical patterns in
plants.
Parastichies: spirals formed by the patterns of
seeds, leaves, scales, etc.
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Observation: number of spirals in each direction.
sunflower head
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Observation: number of spirals in each direction.
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Observation: number of spirals in each direction.
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34 counter-clockwise spirals
55 clockwise spirals
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Parastichies numbers
= (34,55).
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Other types of sunflowers may have numbers
(13,21), or (21,34), or (55,89), or (89,144), …
Pine cones have numbers (2,3), (3,5), (5,8),…
Dragon fruits have numbers (3,5)
Pineapples have numbers (8,13)
Broccoli and cauliflower have numbers (5,8)
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Striking fact: Most parastichies numbers are
consecutive pairs from the Fibonacci
sequence
1, 2, 3, 5, 8,13, 21, 34, 55, 89, 144, ...
(Rarely happened) exceptional cases: some
parastichies numbers are consecutive pairs
from the sequence
2, 4, 6, 10, 16, 26, 42, …
or from the anomalous sequence or Lucas
sequence
1, 3, 4, 7, 11, 18, 29, ...
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Some early history
A third set of spirals: 21 spirals
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triplet of parastichies numbers: (21,34,55)
Ancient Egyptians probably observed and knew about
some regular patterns in plants.
Ancient Greek scholars wrote about regular leaf and plant
patterns.
1400’s: Leonardo da Vinci wrote in a
notebook about parastichy patterns in plants.
1700’s: Charles Bonnet and G.L. Calandrini studied
parastichies of fir cones.
1830‘s: Alexander Braun observed that pairs of
parastichies numbers for pine cones are usually
consecutive Fibonacci numbers.
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Fibonacci numbers
„
Fibonacci found solution to a
problem of monthly
population growth of rabbits.
Leonardo Pisano
(Fibonacci) 1170-1250
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As fn = fní1+ fní2 , the sequence f1 , f2 , f3 , f4 , f5 , … is
1, 2, 3, 5, 8,13, 21, 34, ….
This is called the Fibonacci sequence
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Ratio of consecutive Fibonacci numbers
the ratio
Finding exact value of
I
fn1
fn 2 fn1 fn
f
1 n
and limit of
fn
fn1
fn1
fn1
f
are both I . Taking limits of n1 and
as I ,
f
n
we have
fn 1
, as n becomes bigger and bigger,
fn
Limit of
approaches a particular value I :
2/1 = 2.0
34/21 = 1.619047...
3/2 = 1.5
55/34 = 1.6176470...
5/3 = 1.6666…
89/55 = 1.6181818...
8/5 = 1.6
144/89 = 1.6179775...
13/8 = 1.625
233/144 = 1.61805555...
21/13 = 1.615384…
…...
I 1
1
I
Therefore
, which gives
I
I 2 I 1 0
1 5
1.61803398 ...
2
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„
I
Relationship between phyllotaxis and
Fibonacci numbers
is called the golden ratio.
Interpretations of
I
I
I:
C A
A B
1
I 1
13
I
1
I 1
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Fibonacci numbers
1,2,3,5,8,13,21,...
Golden ratio
Parastiches (spiral)
numbers in plants
?????
?????
I
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Biology: actual growth of sunflower head
or similar plants
primordia
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Generative spiral: continuous spiral through
consecutively formed primordia.
Divergence angle: angle between two consecutive
primordia.
Apex
Generative spiral
primordia
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Around 1837, Auguste and Louis Bravais
discovered that in many cases the divergence
angle is close to 137.5o.
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Another way to get )
360 o y I
This is a special angle !
360 o u I
360 o 222.49223 ...o
137.50776...o
582.49223 ...o
720 o 582.49223 ...o
„
222.49223 ...o
137.50776...o
This special angle is called the golden angle
:
)
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)
: : ) = I : 1 = 1.618...
= 1: I –1
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Relationship between phyllotaxis and Fibonacci numbers
Fibonacci numbers
1,2,3,5,8,13,21,...
Parastichies (spiral)
numbers in plants
Why nature chooses
)
Golden ratio
Golden angle
Hypothetical cases
o
„ case 1: divergence angle (d) = 120
I
)
Divergence
angle
as divergence angle?
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Hypothetical cases
o
o
„ case 2: d = 135 = 360 x (3/8)
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Hypothetical cases
„ All cases in which
d = 360o x (p/q)
where p, q are positive integers (may
assume p and q have no common factors)
will result in q radial lines of seeds.
NO GOOD !
„
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Notice that
p
are rational numbers.
q
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Rational numbers are those numbers in the
form of p where p, q are integers, q nonzero.
„
q
„
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Irrational numbers are all those real numbers
which are not rational.
2 is irrational.
p
Proof Suppose 2 = q . We may assume p, q have no
2
2
2
common factors. Since 2q = p , 2 divides p . Hence 2
divides p. Let p=2r. Then 2q2 = p2 = (2r)2 = 4r2, or q2 = 2r2.
Hence 2 divides q2, and thus 2 divides q. This means p
and q have common factor 2 ---- a contradiction!
Another example of irrational number is log105.
Again we use Proof by Contradiction to show
that log105 is irrational in the following.
Suppose that log105 = p/q where p, q are
integers with q Ћ0. We may assume without
loss of generality that q > 0. Recall that if
logab = c then b = ac. Thus we have 5 = 10p/q,
which gives 5q = 10p. However, 5q is a number
the last digit of which is 5, while 10p is not.
This is a contradiction.
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§
¨
©
o
When d ) ¨360 u
How about the case d = 360o x s
where s is an irrational number ?
„ Hypothetical
case:
s= 2
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·
3 5
137.50...o ¸¸
2
¹
Most tightly
packed
parastichies
numbers: (34,55)
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1979 H. Vogel
observed that
when d = 137.3o
When d = 137.6o
Gaps appear.
No good !
Also no good !
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What is the mathematical properties of I or )
that make them so special to give the most
efficient packing?
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I is irrational, and can be written as a
continued fraction:
I 1
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„
I
can be approximated by fractions of the form:
1
1
1
1
1
1
1
1
1...
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
...
which are 2 , 3 , 5 , 8 , … They are ratios of
1
2
3
5
consecutive pairs of Fibonacci numbers.
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„
Fact: every irrational number s can be written as
continued fraction of the form (a0,,a1, … are
integers, a1, a2, … > 0)
1
a0 1
a1 a3 1
a4 ...
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s can be “best” approximated by rational numbers:
a0
1
a0 a1
a0 1
1
a1 a2
1
a0 a1 a2 1
a3
Thus I can be interpreted as the “most irrational”
number.
1
a0 1
Theorem I is the irrational number such that,
among all other irrational numbers, when using its
best rational approximates for approximation, the
resulting errors are biggest.
1
a2 „
„
1
a1 a2 1
a3 1
a4
...
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Conclusions
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Theories on cause of Fibonacci patterns
The golden angle ), when used as the
divergence angles, gives the most efficient
packing of primordia.
The golden ratio I has rich mathematical
properties that make it the best choice for the
irrational number s for divergence angle
d=360o x s
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Question
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How do plants know these things?
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1868 Wilhelm Hofmeister proposed that each
new primordium forms in the largest gap left
by preceding primordia. 1962 Mary and Robert
Snow proposed that a new primordium will
grow whenever a certain minimal amount of
space becomes available at the apex.
1913 Johannes Schoute argued that primordia
release chemical inhibitors to keep others from
growing too close.
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An experiment on magnetic fluid
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1992, done by S.Douady and Y.Couder.
Tiny drops of magnetic fluid fall at regular
intervals into the center of a dish which was
filled with silicone oil.
The drops, polarized by magnetic field, repel
each other.
Drops are attracted towards edge of dish by
magnetic field.
Sketch of experiment apparatus
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Experiment results
Experiment results
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Patterns formed depend on length of intervals
between drops.
Prevalent patterns are those with a divergence
angle very close to )
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Divergence angles that are not close to )
also occur.
Parastichies numbers from the anomalous
sequence 3, 4, 7, 11, 18, … also occur.
These mean real life exceptional phyllotaxis
cases were reproduced!
Computer simulations, using different
repulsive energy functions for the drops,
were carried out. Results were qualitatively
the same.
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Two roles of mathematics in
research (of other disciplines)
Implication of results
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Mathematical modeling
Research problem
Phenomenon of phyllotaxis patterns
may be results of simple physical or
mathematical laws.
Mathematical model
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Mathematical theory
Research problem
Mathematical theory
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References
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Phyllotaxis, Rober V. Jean, Cambridge University
Press, 1994.
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Life’s Other Secret, Ian Stewart, Allen Lane The
Penguen Press, 1998.
Mathematics in Nature, John A. Adam, Princeton
University Press, 2003.
Phyllotaxis as a physical self-organized growth
process, S. Douady and Y. Couder, Physical Review
Letters, Vol. 68, no. 12, 1992, 2098-2101.
www resources (in “links” section of course website).
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