The icon means there is a Things to do investigation at the end of the section.
On the first page on the Fibonacci Numbers and Nature
we saw that the Fibonacci numbers appeared in (idealised) rabbit, cow and bee populations, and in
the arrangements of petals round a flower, leaves round branches and seeds on
seed-heads and pinecones and in everyday fruit and vegetables.
We explained why they appear in the rabbit, cow and bee populations but what about
the other appearances that we see around us in nature? The answer relates to
why Phi appears so often in plants and the Fibonacci numbers appear because the eye
"sees" the Fibonaci numbers in the spirals of seedheads, leaf arrangements and so on,
and we looked at this on the previous Fibonacci Numbers in Nature page.
So we ask...
Why does nature like using Phi in so many plants?The answer lies in packings - the best arrangement of objects to minimise wasted space.
...square objects would pack most closely in a square array,
whereas round objects pack better in a hexagonal arrangement.... |
What nature seems to use is the same pattern to place seeds on a seedhead as it used to arrange petals around the edge of a flower AND to place leaves round a stem. What is more, ALL of these maintain their efficiency as the plant continues to grow and that's a lot to ask of a single process!
So just how do plants grow to maintain this optimality of design?
Also, these cells grow in a spiral fashion, as if the stem turns by an angle and then a new cell appears, turning again and then another new cell is formed and so on.
These cells may then become a new branch, or perhaps on a flower become petals and stamens.
The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. So, once an angle is fixed for a leaf, say, that leaf will least obscure the leaves below and be least obscured by any future leaves above it. Similarly, once a seed is positioned on a seedhead, the seed continues out in a straight line pushed out by other new seeds, but retaining the original angle on the seedhead. No matter how large the seedhead, the seeds will always be packed uniformly on the seedhead.
And all this can be done with a single fixed angle of rotation between new cells?
Yes! This was suspected by people as early as the last century. The principle that a single
angle produces uniform packings no matter how much growth appears after it was only proved
mathematically in 1993 by Douady and Couder, two french mathematicians.
You will have already guessed what the fixed angle of turn is - it is Phi cells per turn or phi turns per new cell.
Let's now try and show just why phi is the best angle to use in the next few sections of this page.
The links in this section are to Quicktime animations. They are worth viewing as they show the dynamics of what might happen if seeds were not placed with a phi-angle between them.
Why not 0·6 of a turn per seed or 0·5 or 0·48 or 1·6 or some other number?First we can agree that turning 0·6 of a turn is exactly the same as turning 1·6 turns or 2·6 turns or even 12·6 turns because the position of the point looks the same. So we can ignore the whole number part of a turn and only examine the fractional part.
Also, since a 0·6 of a turn in one direction is the same as 0·4 of a turn in the other, we could limit our investigation to turns which are less than 0·5 too. However sometimes it will be easier to talk of fractions of a turn which are bigger than 0·5 or even that are bigger than 1, but the only important part of the number is the fractional part.
So, in terms of seeds - which develop into fruit - what is a fruit-ful numbers? Which has the best properties as a turning angle for our meristem? It turns out that numbers which are simple fractions are not good choices, as we see in the next section.
A circular seedhead is more compact and would have better mechanical strength and so be better able to withstand wind and heavy rain.
Here is 0·48 of a turn between seeds.
What do you think will happen with 0·6 of a turn
between successive seeds?
Did you expect it to be so different?
Notice how the seeds are not equally spaced, but fairly soon settle
down to 5 "arms". Why?
Because 0·6=3/5 so every 3 turns will have produced exactly 5 seeds
and the sixth seed will be at the same angle as the first, the seventh in the same
(angular) position as the second and so on.
The seeds appearing at every third arm, in turn, round and round the
5 arms. So we count 3-of-the-5 (3/5) to find the next "arm" where a seed will appear.
So what seems to be important is just the fractional part of our seeds-per-turn value and we can ignore the whole number part. There is another value that will give the same animation too. What is it? Well, if we went 0·6 of a turn in the other direction, it is equivalent to going 1-0·6=0·4 of a turn between seeds. So also would be 1·4, 1·4, 3·4 and so on.If we try 1·6 or 2·6 or 3·6 can you see that we will get the same animation since the extra whole turns do not affect where the seeds are placed?
Here's what happens if we have a value closer to phi(0·6180339..), namely 0·61. You'll notice that it is better, but that there are still large gaps between the seeds nearest the centre, so the space is not best used. This is also equivalent to using 1·61, 2·61, etc. and also to 1-0·61=0·39 and therefore to 1·39 and 2·39 and so on.
In fact, any number which can be written as an exact ratio (a
rational number) would not be good as a turn-per-seed
angle.
If we use p/q as our angle-turn-between-successive-turns, then we
will end up with q straight arms, the seeds being placed every p-th
arm. [This explains why 0·6=3/5 has 5 arms and the seeds appear at
every third arm, going round and round.]
So what is a "good" value? One that is NOT an exact ratio
since very large seed heads will eventually end up with seeds in
straight lines.
Numbers which cannot be expressed exactly as a ratio are called
irrational numbers (ir-ratio-nal) and this description applies
to such values as
2, Phi, phi,
e,
pi and any multiple
of them too.
You'll notice that the e(2·71828...) animation has 7 arms since
its turns-per-seed is (two whole turns plus)
0·71828... of a turn, which is a bit more
than 5/7(=0·71428..).
A similar thing happens with pi(3·14159..) since the fraction of a turn
left over after 3 whole turns is 0·14159 and is close to
1/7=0·142857.. . It is a little less, so the "arms" bend in
the opposite direction to that of e's (which were a bit
more than 5/7).
These rational numbers are called rational approximations to
the real number value.
If we take more and more seeds, the spirals alter and we get better
and better approximations to the irrational value.
What is "the best" irrational number?
One that never
settles down to a rational approximation for very long. The
mathematical theory is called CONTINUED FRACTIONS.
The simplest such number is that which is expressed as
P=1+1/(1+1/(1+1/(...) or, its reciprocal p=1/(1+1/(1+1/(...))).
P is just 1+1/P, or P^{2}=P+1.
p is just 1/(1+p) so p^{2}+p=1.
We
will see later that these are just definitions of Phi (P) and phi (p) (and their negatives)!
The exact value of Phi is (5 + 1)/2
and of phi is (5 – 1)/2.
Both are irrational numbers whose rational
approximations are ...
phi: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, ... Phi: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ...
which is why you see the Fibonacci spirals in the seed heads!
1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..
Look out! There's the teacher: 30 metres at 3 o'clock!where the axis of 12 o'clock is straight ahead.
So we need a centre point or origin and a fixed line through it
as an axis to measure the angles from.
A point is then uniquely identified by
its distance r from the origin
and its angle theta measured as a rotation about the origin from the axis.
s | r | theta | x | y |
1 | 1 | 0.61803399 | -0.7373689 | -0.6754903 |
2 | 1.41421356 | 1.23606798 | 0.12363865 | 1.4087986 |
3 | 1.73205081 | 1.85410197 | 1.05384702 | -1.3745568 |
4 | 2 | 2.47213595 | -1.969427 | 0.3483639 |
5 | 2.23606798 | 3.09016994 | 1.8866942 | 1.20016041 |
Click on the thumbnail images or links to open a demonstration in a new window.
2D: | 3D: |
3D Stem: |
1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..
1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..
The technical term for the study of the arrangements of leaves and of seedheads in plants is phyllotaxis.
An important technical paper about phi and its optimal properties
for plant growth can be found in
Phyllotaxis as a self-organised growth process by
Stephane Douady and Yves Couder, pages 341 to 352 in
Growth Patterns in Physical Sciences and Biology,
(editor J M Garcia-Ruiz et al), Plenum press, 1993.
A history of the study of phyllotaxis
by I Adler, D Barabe, R V Jean
in Annals of Botany, 1997, Vol.80, No.3, pp.231-244.
A better way to construct the Sunflower head in
Mathematical Biosciences
volume 44, (1979) pages 145 - 174.
A H Church
On the relation of Phyllotaxis to Mechanical Laws,
published by Williams and Norgat, London 1904.
E E Leppik,
Phyllotaxis, anthotaxis and semataxis Acta
Biotheoretica Vol 14, 1961, pages 1-28.
F J Richards
Phyllotaxis: Its Quantitative Expression and Relation to
growth in the Apex Phil. Trans. Series B Vol 235,
1951, pages 509-564.
D'Arcy W Thompson
On Growth and Form Dover Press 1992.
This is the complete edition! (Click on the title-link for more information and
to order it now.)
There is also
an abridged version from Cambridge University press (more information and order it
on line via the title-link.)
T A Davis,
Fibonacci Numbers for Palm Foliar Spirals Acta
Botanica Neelandica, Vol 19, 1970, pages 236-243.
T A Davis
Why Fibonacci Sequence for Palm Leaf Spirals?,
Fibonacci Quarterly, Vol 9, 1971, pages 237-244.
The Algorithmic Beauty of Plants by P
Prusinkiewicz, and A Lindenmayer, published by
Springer-Verlag (Second printing 1996) is an astounding book of wonderful
images and patterns in plant shapes as well as algorithms for modelling and
simulation by computer.
(For more information and how to order it online use the title-link).
Related to this book is:
The Algorithmic Beauty of Sea Shells (Virtual Laboratory) in hardback by
Hans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler (more information and
order it online via this title-link).
The Curves of Life: Being an Account of Spiral Formations and Their Application
to Growth in Nature, to Science, and to Art Sir Theodore A Cook, Dover books, 1979, ISBN 0 486
23701 X.
A Dover reprint of a classic 1914 book. (More information and you can order it
online via the title-link.)
Also see H S M
Coxeter's
Introduction to Geometry, published by Wiley, in its
Wiley Classics Library series,
1989, ISBN 0471504580, especially chapter 11 on
Phyllotaxis. (More information and order it online via the title-link.)
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© 1996-2009 Dr Ron Knott updated 12 February 2009