Fibonacci Numbers In Nature Essay Ralph

Fibonacci Numbers and Nature - Part 2
Why is the Golden section the "best" arrangement?

Contents of this page

The icon means there is a Things to do section of questions to start your own investigations.

On the first page on the Fibonacci Numbers and Naturewe 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 onseed-heads and pinecones and in everyday fruit and vegetables.
We explained why they appear in the rabbit, cow and bee populations but what aboutthe other appearances that we see around us in nature? The answer relates towhy 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.

Packings

If you were asked what was the best way to pack objects your answer woulddepend on the shape of the objects since....
...square objects would pack most closely in a square array,

whereas round objects pack better in a hexagonal arrangement....

So why doesn't nature use one of these? Seeds are round (mostly), so why don't wesee hexagonal arrangments on seedheads?
Although hexagonal symmetry IS the best packing for circular seeds, it doesn't answerthe question of how leaves should be arranged round a stem or how to pack flower-heads(which are circular because that is the shape that encloses maximum area for minimum edge) with seeds that grow in size.

What nature seems to use is the same pattern to place seeds on a seedheadas it used to arrangepetals around the edge of a flower AND to place leaves round a stem. What is more, ALL of these maintaintheir efficiency as the plant continues to grow and that's a lot to askof a single process!

So just how do plants grow to maintain this optimality of design?

The Meristem and Spiral growth patterns

Botanists have shown that plants grow from a single tiny group of cells right atthe tip of any growing plant, called the meristem.There is a separate meristem at the end of each branch or twig where new cells are formed. Once formed, they grow in size, but new cells areonly formed at such growing points. Cells earlier down the stem expand and so thegrowing point rises.

Also, these cells grow in a spiral fashion, as if the stem turns by an angle and thena 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 nomatter 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 leavesabove it. Similarly, once a seed is positioned on a seedhead, the seed continues out ina 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 uniformlyon 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 singleangle produces uniform packings no matter how much growth appears after it was only provedmathematically 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.

Why does the Golden Ratio (Phi) appear in plants?

The arrangements of leaves is the same as for seeds and petals. All are placed at 0·618034.. leaves, (seeds, petals) per turn. In terms of degrees this is0·618034 of 360° which is 222·492...°.However, we tend to "see" the smaller angle which is(1-0·618034)x360 = 0·381966x360 = 137·50776..°.When we look at properties of Phi and phi on a later page, we shall see that

1-phi = phi2 = Phi-2


If there are Phi (1·618...) leaves per turn (or, equivalently, phi=0·618... turns per leaf ), then we have the best packing so that each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible area exposed to falling rain so the rain is directed back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination.
The whole of the plant seems to produce its leaves, flowerhead petalsand then seeds based upon the golden number.

And why do the Fibonacci numbers appear as leaf arrangements and as the number of spirals on seedheads?


The Fibonacci numbers form the best whole number approximations to the golden number, which we examined in greater detail on the first Fibonacci in Nature page.

 

Let's now try and show just why phi is the best angle to use in the next few sections ofthis page.

Why is the Golden section the "best" number?

The links in this section are to Quicktime animations. Theyare worth viewing as they show the dynamics of what might happen ifseeds 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 othernumber?

First we can agree that turning 0·6 of a turn is exactly the same as turning 1·6 turnsor 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, wecould limit our investigation to turns which are less than 0·5 too. However sometimesit will be easier to talk of fractions of a turn which are bigger than 0·5 or even thatare bigger than 1, but the only important part of the number is the fractionalpart.

So, in terms of seeds - which develop into fruit - what is a fruit-ful numbers? Which has thebest properties as a turning angle for our meristem? It turns out thatnumbers which are simple fractions are not good choices, as we see in the next section.

Why exact fractions are fruitless!

Let's first see what happens with a simple number such as 0·5 turns per seed.
Since 0·5=1/2 we get just 2 "arms" andthe seeds use the space on the seedhead very inefficiently:the seedhead is long and floppy. The picture is a link to an animation where you cansee the new seeds appearing at the centre as the older ones continue growing outwardsin a straight line from the central growing point (where the new seed cells appear).

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.
[The picture is again a link to an animation.]
The seeds seem to be sprayed from two revolving"arms". This is because 0·48 is very close to 0·5 and a half-turnbetween seeds would mean that they would just appear on alternatesides, in a straight line. Since 0·48 is a bit less than 0·5,the "arms" seem to rotate backwards a bit each time.
So if we has 0·52 seeds per turn, we would be a little in advance of half a turn and the final pattern would be a mirror-image (as if we had used 1-0·52=0·48 seeds per turnbut turning in the opposite direction).


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 settledown to 5 "arms". Why?
Because 0·6=3/5 so every 3 turns will have produced exactly 5 seedsand 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 the5 arms. So we count 3-of-the-5 (3/5) to find the next "arm" where a seed will appear.

If we try 1·6 or 2·6 or 3·6 can you see thatwe will get the same animation since the extra whole turns do notaffect where the seeds are placed?

So what seems to be important is just the fractional partof 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 isit? Well, if we went 0·6 of a turn in the other direction, it isequivalent to going 1-0·6=0·4 of a turn between seeds. So also wouldbe 1·4, 1·4, 3·4 and so on.

Here's what happens if we have a value closer tophi(0·6180339..), namely0·61. You'll notice thatit is better, but that there are still large gaps between the seedsnearest the centre, so the space is not best used. This is alsoequivalent to using 1·61, 2·61, etc. and also to 1-0·61=0·39 andtherefore to 1·39and 2·39 and so on.

In fact, any number which can be written as an exact ratio (arational number) would not be good as a turn-per-seedangle.
If we use p/q as our angle-turn-between-successive-turns, then wewill end up with q straight arms, the seeds being placed every p-tharm. [This explains why 0·6=3/5 has 5 arms and the seeds appear atevery third arm, going round and round.]

The rational answer is an irrational number!

So what is a "good" value? One that is NOT an exact ratiosince very large seed heads will eventually end up with seeds instraight lines.
Numbers which cannot be expressed exactly as a ratio are calledirrational numbers (ir-ratio-nal) and this description appliesto such values as 2, Phi, phi,e,pi and any multipleof them too.

You'll notice that the e(2·71828...) animation has 7 arms sinceits turns-per-seed is (two whole turns plus)0·71828... of a turn, which is a bit morethan 5/7(=0·71428..).
A similar thing happens with pi(3·14159..) since the fraction of a turnleft over after 3 whole turns is 0·14159 and is close to1/7=0·142857.. . It is a little less, so the "arms" bend inthe opposite direction to that of e's (which were a bitmore than 5/7).
These rational numbers are called rational approximations tothe real number value.
If we take more and more seeds, the spirals alter and we get betterand better approximations to the irrational value.

What is "the best" irrational number?
One that neversettles down to a rational approximation for very long. Themathematical theory is called CONTINUED FRACTIONS.
The simplest such number is that which is expressed asP=1+1/(1+1/(1+1/(...) or, its reciprocal p=1/(1+1/(1+1/(...))).
P is just 1+1/P, or P2=P+1.
p is just 1/(1+p) so p2+p=1.
Wewill 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 rationalapproximations 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..

Interactive Demonstrations

How the demonstrations are made

In all the following interactive demonstrations, the following principles are used in the programming:
  1. The seeds are given numbers from 1 up as far as you want to go.
  2. The position of a seed s given in polar coordinates which involve two numbers:
    • a distance, r, measured from a fixed point called the origin and
    • an angle of rotation, theta, measured from a fixed direction called the axis
    We use this system in everyday language when we say:
    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.

    1. The distance r of seed number s is √s,
      so for example, seed number s = 9 is at a distance 3 from the origin
    2. The angle theta that seed s has as seen from the origin is s × phi of a whole turn, where phi = 0.6180339... = (√5 – 1 ) / 2
      so for example, seed s = 9 is 9 phi = 9 × 0.61804 = 5.5624 turns.
      Since a whole turn takes us back to the starting direction, we can ignore the whole-number part and all we need is the fractional part 0.5624 of a turn but it does no harm to use 5.5624 either.
  3. To plot the seedhead:
    Most computer languages use a graphing system with x and y axes called cartesian coordinates.
    They also provide functions on angles where the unit of angle is a radian that is, an angle is measured by the distance it makes around a circle of unit radius.
    Because a unit circle has a circumference of 2π, there are 2π = 2 3.14159267... = 6.2831853... radians in one complete turn.
    Now we can convert the distance-angle coordinates for a seed into cartesian coordinates as follows:
    1. The x-coordinate is r cos( 2 π theta)
    2. The y-coordinate is r sin( 2 π theta )
Here is a check on the values for the first 5 seeds:
srthetaxy
110.61803399-0.7373689-0.6754903
21.414213561.236067980.123638651.4087986
31.732050811.854101971.05384702-1.3745568
422.47213595-1.9694270.3483639
52.236067983.090169941.88669421.20016041
Now plot your seeds on an x-y graph using the cartesian coordinates for each seed.
By multiplying the seed number by a different ratio to get its angle, you can experiment and see what seed heads would be like if nature used a different value for its turns-per-seed value too.

Click on the thumbnail images or links to open a demonstration in a new window.

Quicktime

Here is another Quicktime movie which shows various turns-per-seed values near phi (0·61803) showing that there are always gapstowards the outer edge of the "seedhead"and that phi gives the best value for all sizes of flowerhead.
Depending on your browser, you should be able to move the slider to any part of the movie to view individual frames and values near phi.

Excel Spreadsheet

This Excel spreadsheet employs a slider to alter the turns-per-seed on an interactive chartto show that 0.61804 is better than0.61803 or 0.61805 in terms of the evenness of thedistribution of 2000 seeds on a seedhead. A smaller view of the innermost 50 seedsis given as well. It does not use Macros so disable them if asked when you load it.

Mathematica

There are some nice interactive phyllotaxis demonstrationsmade with Mathematica from Wolfram Research in their Demonstrations Project.
Each can either be viewed without Mathematicain a browser page (as a movie of the demonstration)or by downloading the free Mathematica Playerin which case the demonstration is fully interactive.

Maple

Waterloo Software's Mapleis another professional mathematics package similar to Mathematica.The Maple code here illustrates the algorithm used to generate adiagram of seeds with a given number of turns-per-seed.
The Maple code:
> with(plots):
> growpts1 := proc(n, TurnperSeed, symb)
local i, a, r, s, phi2pi;
s := null; phi2pi := 2*TurnperSeed*Pi; listplot(
[seq([sqrt(n - i)*cos(phi2pi*i), sqrt(n - i)*sin(phi2pi*i)], i = (1 .. n))],
style = POINT, axes = NONE, scaling = CONSTRAINED, symbol = symb);
end proc:
> growpts:=(n,TpS)->growpts1(n,TpS,POINT):
> growpts(1000,Pi);


> seedplot := proc(n, ratio)
display([seq(growpts1(i, ratio, CIRCLE), i = (1 .. n))], insequence = true,
style = point )
end proc;
seedplot := proc(n, ratio)
display([seq(growpts1(i, ratio, CIRCLE), i = (1 .. n))], insequence = true,
style = point)
end proc;
> display([seq(growpts1(i, (sqrt(5)-1)/2, CIRCLE), i = (1 .. 10))], insequence = true);

Geometer's Sketchpad

Try this Geometer's Sketchpadactive demonstration which lets you alter the inter-seedangle at will (and animate it) to see just why the golden section angle produces the bestpacking.
Geometer's Sketchpad is available as a free trial for PC and Apple Macfor teachers and administrators.

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

You do the maths...

  1. The "rational approximations" to real numbers are better seen if, instead of producing seeds at the centre, we keep adding them round the outside - that is, along the square-root spiral which has equation R=A where R is the (radial) distance of a point from the origin, and A its angle turn (from the 0 angle direction). Use any of the Demonstrations to "grow plants" that will find good rational approximations to a decimal fraction of your choice. For example, Pi as the angle of rotation between seeds, shows 7 arms clearly after only 100 seeds, gets confused at about 500 seeds but by 1000 shows a better approximation - there are 113 "arms", seeds being grown every 16 showing that a better approximation for Pi is 3+16/113=355/113. [As Jordi Mas pointed out to me, this approximation for pi was known in China as far back as the year 500!]
  2. What about approximations to sqrt(3) or sqrt(5)?
  3. Take sqrt(3) and plot lots of "seeds".
    What sequence of approximations do you get? You should be able to answer this if you plot 500 seeds.
  4. Now convert each approximation into a continued fraction. What pattern in the numbers in the continued fraction emerges?
  5. Try to prove that the pattern continues indefinitely, by proving its value is sqrt(3).

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

Links and References

Phyllotaxis

The technical term for the study of the arrangements of leaves andof 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.

Fibonacci Numbers in Nature

Here are some not-too technical papers about the maths whichjustifies the occurrence of the Fibonacci numbers in nature:
  • A H Church On the relation of Phyllotaxis to Mechanical Laws,published by Williams and Norgat, London 1904.
  • Phyllotaxis, anthotaxis and semataxis ActaBiotheoretica Vol 14, 1961, pages 1-28.
  • Phyllotaxis: Its Quantitative Expression and Relation togrowth in the Apex Phil. Trans. Series B Vol 235,1951, pages 509-564.
  • D'Arcy W ThompsonOn Growth and Form Dover Press 1992.
    This is the complete edition! (Click on the title-link for more information andto order it now.)
    There is also an abridged version from Cambridge University press (more information and order iton line via the title-link.)
  • T A Davis, Fibonacci Numbers for Palm Foliar Spirals ActaBotanica 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 PPrusinkiewicz, and A Lindenmayer, published bySpringer-Verlag (Second printing 1996) is an astounding book of wonderfulimages and patterns in plant shapes as well as algorithms for modelling andsimulation 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 byHans Meinhardt, Przemyslaw Prusinkiewicz, Deborah R. Fowler (more information andorder 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 48623701 X.
    A Dover reprint of a classic 1914 book. (More information and you can order itonline via the title-link.)
  • Also see H S MCoxeter'sIntroduction to Geometry, published by Wiley, in itsWiley Classics Library series,1989, ISBN 0471504580, especially chapter 11 onPhyllotaxis. (More information and order it online via the title-link.)

WWW Links

  • Eric W. Weisstein's page on Phyllotaxis has some more references tobooks and articles.
  • Eddy Levin has invented a wonderful golden-section measuring tool, like a pair ofdividers or callipers and he has a page of examples of it in use showing thegolden section on flowers, insects, leaves etc that's well worth looking at. Click on his"Dental" link and you can see that, as a dentist, he sees the golden section every dayin the arrangement and width of human teeth too!

© 1996-2016 Dr Ron Knott
updated 23 June 2017

Fibonacci Numbers and Nature

This page has been split into TWO PARTS.

This, the first, looks at the Fibonaccinumbers and why they appear in various "family trees" and patterns of spiralsof leaves and seeds.

The second page then examines why the golden section is used by nature in some detail, including animations of growing plants.

Contents of this page

The icon means there is a You do the maths... section of questions to start your own investigations.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Rabbits, Cows and Bees Family Trees

Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at twoadaptations of it to make it more realistic. This introduces you to the Fibonacci Numberseries and the simple definition of the whole never-ending series.

Fibonacci's Rabbits

The original problem that Fibonacci investigated (in the year1202) was about how fast rabbits could breed in ideal circumstances.

Suppose a newly-born pair of rabbits, one male, one female, areput in a field. Rabbits are able to mate at the age of one month so thatat the end of its second month a female can produce another pair ofrabbits. Suppose that our rabbits never die and thatthe female always produces one new pair (one male,one female) every month from the second month on. The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

  1. At the end of the first month, they mate, but there is still one only 1 pair.
  2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
  3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
  4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.



The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Can you see how the series is formed and how it continues? If not,look at the answer!

The first 300 Fibonaccinumbers are here and some questions for you to answer.

Now can you see why this is the answer to ourRabbits problem? If not, here's why.
Another view of the Rabbit's Family Tree:

Both diagrams above represent the same information. Rabbits have been numbered to enablecomparisons and to count them, as follows:
  • All the rabbits born in the same month are of the same generation and are on the same level in the tree.
  • The rabbits have been uniquely numbered so that in the same generation the new rabbits are numbered in the order of their parent's number. Thus 5, 6 and 7 are the children of 0, 1 and 2 respectively.
  • The rabbits labelled with a Fibonacci number are the children of the original rabbit (0) at the top of the tree.
  • There are a Fibonacci number of new rabbits in each generation, marked with a dot.
  • There are a Fibonacci number of rabbits in total from the top down to any single generation.
There are many other interesting mathematical properties of this tree that are explored in later pagesat this site.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

The Rabbits problem is not very realistic, is it?

It seems to imply that brother and sisters mate, which,genetically, leads to problems. We can get round this by saying thatthe female of each pair mates with any male and produces anotherpair.
Another problem which again is not true to life, is that each birth is ofexactly two rabbits, one male and one female.

Dudeney's Cows

The English puzzlist, Henry E Dudeney (1857 - 1930, pronounced Dude-knee)wrote several excellent books of puzzles (see after this section). In one of them he adapts Fibonacci'sRabbits to cows, making the problem more realistic in the way we observed above.He gets round the problems by noticing that really, it is only the females that are interesting - er - I mean the number of females!

He changes months intoyears and rabbits into bulls (male) and cows (females) in problem 175 in his book536 puzzles and Curious Problems (1967, Souvenir press):

If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?
This is a better simplification of the problem and quite realistic now.

But Fibonacci does what mathematicians often do at first,simplify the problem and see what happens - and the series bearinghis name does have lots of other interesting and practicalapplications as we see later.
So let's look at another real-life situation that is exactly modelled by Fibonacci's series - honeybees.

Puzzle books by Henry E Dudeney

Amusements in Mathematics, Dover Press, 1958, 250 pages.
Still in print thanks to Dover in a very sturdy paperback format at an incredibly inexpensive price. This is a wonderful collection that I find I often dip into. There are arithmetic puzzles, geometric puzzles, chessboard puzzles, an excellent chapter on all kinds of mazes and solving them, magic squares, river crossing puzzles, and more, all with full solutions and often extra notes! Highly recommended!

536 Puzzles and Curious Problems is now out of print, but you may be able to pick up a second hand version by clicking on this link. It is another collection like Amusements in Mathematics (above) but containing different puzzles arranged in sections: Arithmetical and Algebraic puzzles, Geometrical puzzles, Combinatorial and Topological puzzles, Game puzzles, Domino puzzles, match puzzles and "unclassified" puzzles. Full solutions and index. A real treasure.

The Canterbury Puzzles, Dover 2002, 256 pages. More puzzles (not in the previous books) the first section with some characters from Chaucer's Canterbury Tales and other sections on the Monks of Riddlewell, the squire's Christmas party, the Professors puzzles and so on and all with full solutions of course!

Honeybees and Family trees

There are over 30,000 species of bees and in most of them the bees livesolitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact,there are many unusual features of honeybees and in this section we will showhow the Fibonacci numbers count a honeybee's ancestors (in this section a "bee" will mean a "honeybee").
First, someunusual facts about honeybees such as: not all of them have twoparents!
In acolony of honeybees there is one special female called thequeen.
There are many worker bees who are female too but unlike the queen bee, they produce no eggs.
There are some drone bees who are male and do no work.
Males are produced by the queen's unfertilised eggs, so male bees only have a mother but no father!
All the females are produced when the queen has mated with a male and sohave two parents. Females usually end up as worker bees but some arefed with a special substance called royal jelly which makesthem grow into queens ready to go off to start a new colony when thebees form a swarm and leave their home (ahive) in search of a place to build a new nest.

So female bees have 2 parents, a male and a female whereas malebees have just one parent, a female.

Here we follow the convention of Family Treesthat parents appear above their children, so the latest generations areat the bottom and the higher up we go, the older people are. Such treesshow all the ancestors (predecessors, forebears, antecedents) of the personat the bottom of the diagram. We would get quite a different tree if we listed all the descendants(progeny, offspring) of a personas we did in the rabbit problem, where we showed all the descendants of the original pair.

Let's look at the family tree of a male drone bee.

  1. He had 1 parent, a female.
  2. He has 2 grand-parents, since his mother had two parents, a male and a female.
  3. He has 3 great-grand-parents: his grand-mother had two parents but his grand-father had only one.
  4. How many great-great-grand parents did he have?

Again we see the Fibonacci numbers :

great- great,great gt,gt,gt grand- grand- grand grandNumber of parents: parents: parents: parents: parents:of a MALE bee: 1 2 3 5 8of a FEMALE bee: 2 3 5 8 13 The Fibonacci Sequence as it appears inNature by S.L.Basin in Fibonacci Quarterly, vol 1 (1963), pages 53 - 57.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

You do the maths...

  1. Make a diagram of your own family tree. Ask your parents and grandparents and older relatives as each will be able to tell you about particular parts of your family tree that other's didn't know. It can be quite fun trying to see how far back you can go. If you have them put old photographs of relatives on a big chart of your Tree (or use photocopies of the photographs if your relatives want to keep the originals). If you like, include the year and place of birth and death and also the dates of any marriages.
  2. A brother or sister is the name for someone who has the same two parents as yourself. What is a half-brother and half-sister?
    Describe a cousin but use simpler words such as brother, sister, parent, child?
    Do the same for nephew and niece. What is a second cousin? What do we mean by a brother-in-law, sister-in-law, mother-in-law, etc? Grand- and great- refer to relatives or your parents. Thus a grand-father is a father of a parent of yours and great-aunt or grand-aunt is the name given to an aunt of your parent's.

    Make a diagram of Family Tree Names so that "Me" is at the bottom and "Mum" and "Dad" are above you. Mark in "brother", "sister", "uncle", "nephew" and as many other names of (kinds of) relatives that you know. It doesn't matter if you have no brothers or sisters or nephews as the diagram is meant to show the relationships and their names.
    [If you have a friend who speaks a foreign language, ask them what words they use for these relationships.]

  3. What is the name for the wife of a parent's brother?
    Do you use a different name for the sister of your parent's?
    In law these two are sometimes distinguished because one is a blood relative of yours and the other is not, just a relative through marriage.
    Which do you think is the blood relative and which the relation because of marriage?
  4. How many parents does everyone have?
    So how many grand-parents will you have to make spaces for in your Family tree?
    Each of them also had two parents so how many great-grand-parents of yours will there be in your Tree?
    ..and how many great-great-grandparents?
    What is the pattern in this series of numbers?
    If you go back one generation to your parents, and two to your grand-parents, how many entries will there be 5 generations ago in your Tree? and how many 10 generations ago?

    The Family Tree of humans involves a different sequence to the Fibonacci Numbers. What is this sequence called?

  5. Looking at your answers to the previous question, your friend Dee Duckshun says to you:
    • You have 2 parents.
    • They each have two parents, so that's 4 grand-parents you've got.
    • They also had two parents each making 8 great-grand-parents in total ...
    • ... and 16 great-great-grand-parents ...
    • ... and so on.
    • So the farther back you go in your Family Tree the more people there are.
    • It is the same for the Family Tree of everyone alive in the world today.
    • It shows that the farther back in time we go, the more people there must have been.
    • So it is a logical deduction that the population of the world must be getting smaller and smaller as time goes on!
    Is there an error in Dee's argument? If so, what is it? Ask your maths teacher or a parent if you are not sure of the answer!

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Fibonacci numbers and the Golden Ratio

If we take the ratio of two successive numbers in Fibonacci'sseries, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:

1/1 = 1,  2/1 = 2,  3/2 = 1·5,  5/3 = 1·666...,  8/5 = 1·6,  13/8 = 1·625,  21/13 = 1·61538...

It is easier to see what is happening if we plot the ratios on agraph:

The ratio seems to be settling down to a particular value, whichwe call the golden ratio or the golden number.It has a value of approximately1·618034 , although we shall find an even more accurate value on a later page [this link opens a new window] .

You do the maths...

  1. What happens if we take the ratios the other way round i.e. we divide each number by the one following it: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ..?
    Use your calculator and perhaps plot a graph of these ratios and see if anything similar is happening compared with the graph above.
    You'll have spotted a fundamental property of this ratio when you find the limiting value of the new series!

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The golden ratio 1·618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a Greek letter Phi. The closely related value which we write as phi with a small "p"is just the decimal part of Phi, namely 0·618034.

Fibonacci Rectangles and Shell Spirals

We can make another picture showing the Fibonacci numbers1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1next to each other. On top of both of these draw a square of size 2(=1+1).

We can now draw a new square - touching both a unit square and thelatest square of side 2 - so having sides 3 units long; and thenanother touching both the 2-square and the 3-square (which has sidesof 5 units). We can continue adding squares around the picture,each new square having a side which is as long as the sum of thelatest two square's sides. This set of rectangles whose sides aretwo successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we will call theFibonacci Rectangles.

Here is a spiral drawn in the squares, a quarter of a circle in each square.The spiral is not a truemathematical spiral (since it is made up of fragments which are parts of circles and does not go ongetting smaller and smaller)but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals are seen in the shape of shells of snails and sea shells and, as we see later, in the arrangement of seeds on flowering plants too.The spiral-in-the-squares makes a line from the centre of the spiral increase by a factor ofthe golden number in each square. So points on the spiral are 1.618 times as far from the centreafter a quarter-turn. In a whole turn the points on a radius out from the centreare 1.6184 = 6.854 times further out than when the curve last crossed the same radial line.

Cundy and Rollett (Mathematical Models, second edition 1961, page 70) say that this spiral occurs in snail-shells and flower-headsreferring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6 "The Equiangular Spiral". Here Thompson istalking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phiexpansion factor.

Below are images of cross-sections of a Nautilus sea shell. They show the spiral curve of the shell and the internal chambers that the animal using it adds on as it grows. The chambers providebuoyancy in the water. Click on the picture to enlarge it in a new window.Draw a line from the centre out in any direction and find twoplaces where the shell crosses it so that the shell spiral has gone round just oncebetween them. The outer crossing point will be about 1.6 times as far from the centre as the next inner point on the line where the shell crosses it.This shows that the shell has grown by a factor of the golden ratio in one turn.
On the poster shown here, this factor varies from 1.6 to 1.9 and may be due to the shell notbeing cut exactly along a central plane to produce the cross-section.

Several organisations and companies have a logo based on this design, using the spiral of Fibonacci squares and sometime with the Nautilus shell superimposed.It is incorrect to say this is a Phi-spiral. Firstly the "spiral" is only an approximationas it is made up of separate and distinctquarter-circles; secondly the (true) spiral increases by a factor Phi every quarter-turnso it is more correct to call it a Phi4 spiral.

Click on the logos to find out more about the organisations.

Here are some more postersavailable from AllPosters.com that are great for your study wall or classroomor to go with a science project. Click on the pictures to enlarge them in a new window.

The curve of this shell is called Equiangular or Logarithmic spiralsand are common in nature, though the 'growth factor' may not always be the golden ratio.

  • The Curves of Life Theodore A Cook, Dover books, 1979, ISBN 0 486 23701 X.
    A Dover reprint of a classic 1914 book.

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Fibonacci Numbers, the Golden Section and Plants

One plant in particular shows the Fibonacci numbers in the numberof "growing points" that it has. Suppose that when a plant puts out anew shoot, that shoot has to grow two months before it is strongenough to support branching. If it branches every month after that atthe growing point, we get the picture shown here.

A plant that grows very much like this is the "sneezewort":Achillea ptarmica.

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Flowers, Fruit and Leaves

On many plants, the number of petals is a Fibonacci number:
buttercups have 5 petals; lilies and iris have 3 petals; somedelphiniums have 8; corn marigolds have 13 petals; some asters have21 whereas daisies can be found with 34, 55 or even 89 petals.
The links here are to various flower and plant catalogues:

  • the Dutch Flowerweb's searchable index called Flowerbase.
  • The US Department of Agriculture's Plants Database containing over 1000 images, plant information and searchable database.
3 petals: lily, iris
      Mark Taylor (Australia), a grower of Hemerocallis and Liliums (lilies) points out that although these appear to have 6 petals as shown above, 3 are in fact sepals and 3 are petals. Sepals form the outer protection of the flower when in bud. Mark's Barossa Daylilies web site (opens in a new window) contains many flower pictures where the difference between sepals and petals is clearly visible.
4 petals Very few plants show 4 petals (or sepals) but some, such as the fuchsia above, do. 4 is not a Fibonacci number! We return to this point near the bottom of this page.
5 petals: buttercup, wild rose, larkspur, columbine (aquilegia), pinks (shown above)
      The humble buttercup has been bred into a multi-petalled form.
8 petals: delphiniums
13 petals: ragwort, corn marigold, cineraria, some daisies
21 petals: aster, black-eyed susan, chicory
34 petals: plantain, pyrethrum
55, 89 petals: michaelmas daisies, the asteraceae family.
Some species are very precise about the number of petalsthey have - e.g. buttercups, but others have petals that are very nearthose above, with the average being a Fibonacci number.

Here is a passion flower (passiflora incarnata) from the back and front:

Back view:
the 3 sepals that protected the bud are outermost,
then 5 outer green petals followed by an inner layer of 5 more paler green petals
Front view:
the two sets of 5 green petals are outermost,
with an array of purple-and-white stamens (how many?);
in the centre are 5 greenish stamens (T-shaped) and
uppermost in the centre are 3 deep brown carpels and style branches)

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Seed heads

This poppy seed head has 13 ridges on top.

 

Fibonacci numbers can also be seen in the arrangement ofseeds on flower heads. The picture here is Tim Stone's beautiful photograph of a Coneflower, used here by kind permission of Tim. The part of the flowerin the picture is about 2 cm across.It is a member of the daisy family with the scientific nameEchinacea purpura and native to the Illinois prairie where he lives.

You can have a lookat some more of Tim's wonderfulphotographs on the web.

 

You can see that the orange "petals" seem to form spirals curving both tothe left and to the right. At the edge of the picture, if you count those spiralling to the right as you go outwards, there are 55 spirals. A little further towards the centre and you can count 34 spirals. How many spirals go the other way at these places?You will see that the pair of numbers (counting spirals in curing left and curving right)are neighbours in the Fibonacci series.

Here is a picture of a 1000 seed seedhead with the mathematically closest seeds shownand the closest 3 seeds and a larger seedhead of 3000 seeds with the nearest seedsshown. Each clearly reveals the Fibonacci spirals:
A larger image appears in the book 50 Visions of Mathematics Sam Parc (Editor) published by Oxford and also available for the Kindle.

Click on the picture on the right to see it in more detail in a separate window.

The same happens in many seed and flower heads in nature. The reason seems tobe that this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage,all the seeds being the same size, no crowding in the centre and nottoo sparse at the edges.

The spirals are patterns that the eye sees,"curvier" spirals appearing near the centre, flatter spirals (andmore of them) appearing the farther out we go.

So the number of spirals we see, in either direction, is different forlarger flower heads than for small. On a largeflower head, we see more spirals further out than we do near the centre.The numbers of spirals in each direction are (almost always) neighbouring Fibonacci numbers!Click on these links for some more diagrams of 500, 1000 and 5000 seeds.

Click on theimage on the right for a Quicktime animation of 120 seeds appearingfrom a single central growing point. Each new seed is just phi(0·618) of a turn from the last one (or, equivalently, there are Phi(1·618) seeds per turn). The animation shows that, no matter how bigthe seed head gets, the seeds are always equally spaced. At allstages the Fibonacci Spirals can be seen.

The same pattern shown by these dots (seeds) is followed if thedots then develop into leaves or branches or petals. Each dotonly moves out directly from the central stem in a straight line.

This process models what happens in nature when the "growing tip"produces seeds in a spiral fashion. The only active area is thegrowing tip - the seeds only get bigger once they have appeared.

[This animation was produced by Maple. If there are N seeds in oneframe, then the newest seed appears nearest the central dot, at 0·618of a turn from the angle at which the last appeared. A seed which isi frames "old" still keeps its original angle from the exact centre but will have moved out to a distance which is the square-root of i.]

Phyllotaxis : A Systemic Study in Plant Morphogenesis (Cambridge Studies in Mathematical Biology) by Roger V. Jean (400 pages, Cambridge University Press, 1994) has a good illustrationon its cover - click on the book's title link or this little picture of the coverand on the page that opens, click on picture of the front coverto see it. It clearly shows that the spirals the eye sees are different near the centre on a real sunflowerseed head, with all the seeds the same size.

Smith College (Northampton, Massachusetts, USA)has an excellent website : An Interactive Site for the Mathematical Study of Plant Pattern Formation which is well worth visiting. It also has a page of links to more resources.

Note that you will not always find the Fibonacci numbers inthe number of petals or spirals on seed heads etc., althoughtheyoften come close to the Fibonacci numbers.

You do the maths...

  1. Why not grow your own sunflower from seed?
    I was surprised how easy they are to grow when the one pictured above just appeared in a bowl of bulbs on my patio at home in the North of England. Perhaps it got there from a bird-seed mix I put out last year? Bird-seed mix often has sunflower seeds in it, so you can pick a few out and put them in a pot. Sow them between April and June and keep them warm.
    Alternatively, there are now a dazzling array of colours and shapes of sunflowers to try. A good source for your seed is:Nicky's Seedswho supplies the whole range of flower and vegetable seed including sunflower seed in the UK.
  2. Have a look at the online catalogue at Nicky's Seeds where there are lots of pictures of each of the flowers.
    1. Which plants show Fibonacci spirals on their flowers?
    2. Can you find an example of flowers with 5, 8, 13 or 21 petals?
    3. Are there flowers shown with other numbers of petals which are not Fibonacci numbers?

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Pine cones

Pine cones show the Fibonacci Spirals clearly. Here is a pictureof an ordinary pine cone seen from its base where the stalk connects it to the tree.
Can you see the two sets of spirals?
How many are there in each set?
Here is another pine cone. It is not only smaller, but has a different spiral arrangement.
Use the buttons to help count the number of spirals in each direction] on this pine cone.

You do the maths...

  1. Collect some pine cones for yourself and count the spirals in both directions.
    A tip: Soak the cones in water so that they close up to make counting the spirals easier. Are all the cones identical in that the steep spiral (the one with most spiral arms) goes in the same direction?
  2. What about a pineapple? Can you spot the same spiral pattern? How many spirals are there in each direction?
  • From St. Mary's College (Maryland USA),Professor Susan Goldstine
    has a page with reallygood pine cone picturesshowing the actual order of the open "petals" of the cone numbered down the cone.
  • Fibonacci Statistics in Conifers A Brousseau , The Fibonacci Quarterly vol 7 (1969) pages 525 - 532
    You will occasionally find pine cones that do not have a Fibonacci number ofspirals in one or both directions. Sometimes this is due to deformities producedby disease or pests but sometimes the cones look normal too. This article reports on a study of this question and others in a large collection of Californian pine cones ofdifferent kinds. The author also found that there were as manywith the steep spiral (the one with more arms) going to the left as to the right.
  • Pineapples and Fibonacci Numbers P B Onderdonk The Fibonacci Quarterlyvol 8 (1970), pages 507, 508.
  • On the trail of the California pine, A Brousseau, The Fibonacci Quarterly vol 6 (1968)pages 69-76
    pine cones from a large variety of different pine trees in California were examined andall exhibited 5,8 or 13 spirals.

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Leaf arrangements

Also, many plants show the Fibonacci numbers in thearrangements of the leaves around their stems. If we look down on aplant, the leaves are often arranged so that leaves above do not hideleaves below. This means that each gets a good share of the sunlightand catches the most rain to channel down to the roots as it runsdown the leaf to the stem.
Here's a computer-generated image, based on an African violet type of plant, whereasthis has lots of leaves.

Leaves per turn

The Fibonacci numbers occurwhen counting both the number of times we go around the stem, goingfrom leaf to leaf, as well as counting the leaves we meet until weencounter a leaf directly above the starting one.

If we count in theother direction, we get a different number of turns for the samenumber of leaves.

The number of turns in each direction and thenumber of leaves met are three consecutive Fibonaccinumbers!

For example, in the top plant in the picture above, we have3 clockwise rotations before we meet a leaf directlyabove the first, passing 5 leaves on the way. If wego anti-clockwise, we need only 2 turns. Notice that2, 3 and 5 are consecutive Fibonacci numbers.
For the lower plant in the picture, we have 5clockwise rotations passing 8 leaves, or just3 rotations in the anti-clockwise direction. Thistime 3, 5 and 8 are consecutive numbers in the Fibonacci sequence.
We can write this as, for the top plant, 3/5 clockwiserotations per leaf ( or 2/5 for the anticlockwisedirection). For the second plant it is 5/8 of a turn per leaf (or 3/8).


The sunflower here when viewed from the top shows the same pattern. It is the same plant whose side view is above.Starting at theleaf marked "X", we find the next lower leaf turning clockwise. Numbering the leaves produces the patterns shown here on the right.

The leaves here are numbered in turn, each exactly 0.618 of a clockwise turn (222.5°) from the previous one.
You will see that the third leaf and fifth leaves are next nearest below our starting leaf but the next nearest below it is the 8th then the 13th. How many turns did it take to reach each leaf?
Leaf
number
turns
clockwise
31
52
83
The pattern continues with Fibonacci numbers in each column!

Leaf arrangements of some common plants

One estimate is that 90 percent of allplants exhibit this pattern of leaves involving the Fibonaccinumbers.

Some common trees with their Fibonacci leaf arrangement numbersare:

where t/n means each leaf is t/n of a turn after the last leaf or that there isthere are t turns for n leaves.

Cactus's spines often show the same spirals as we have already seen on pine cones, petals and leaf arrangements, but they are much more clearly visible. Charles Dills has noted thatthe Fibonacci numbers occur in Bromeliads and his Home page has links to lots of pictures.

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Vegetables and Fruit


Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in outline.Looking carefully, you can see a centre point, where the florets are smallest.Look again, and you will see the florets are organised in spirals around this centre in both directions.

How many spirals are there in each direction?

These buttons will show the spirals more clearly for you to count (lines are drawn between the florets):


Romanesque Broccoli/Cauliflower (or Romanesco) looks and tastes like a cross between broccoli and cauliflower. Each floret is peaked and is an identical but smaller version of the whole thing and this makes the spirals easy to see.

How many spirals are there in each direction?

These buttons will show the spirals more clearly for you to count (lines are drawn between the florets):

Here are some investigations to discover the Fibonaccinumbers for yourself in vegetables and fruit.

You do the maths...

  1. Take a look at a cauliflower next time you're preparing one:
    1. First look at it:
      • Count the number of florets in the spirals on your cauliflower. The number in one direction and in the other will be Fibonacci numbers, as we've seen here. Do you get the same numbers as in the picture?
      • Take a closer look at a single floret (break one off near the base of your cauliflower). It is a mini cauliflower with its own little florets all arranged in spirals around a centre.
        If you can, count the spirals in both directions. How many are there?
    2. Then, when cutting off the florets, try this:
      • start at the bottom and take off the largest floret, cutting it off parallel to the main "stem".
      • Find the next on up the stem. It'll be about 0·618 of a turn round (in one direction). Cut it off in the same way.
      • Repeat, as far as you like and..
      • Now look at the stem. Where the florets are rather like a pine cone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the Fibonacci numbers.
  2. Try the same thing for broccoli.
  3. Chinese leaves and lettuce are similar but there is no proper stem for the leaves. Instead, carefully take off the leaves, from the outermost first, noticing that they overlap and there is usually only one that is the outermost each time. You should be able to find some Fibonacci number connections.
  4. Look for the Fibonacci numbers in fruit.
    1. What about a banana? Count how many "flat" surfaces it is made from - is it 3 or perhaps 5? When you've peeled it, cut it in half (as if breaking it in half, not lengthwise) and look again. Surprise! There's a Fibonacci number.
    2. What about an apple? Instead of cutting it from the stalk to the opposite end (where the flower was), i.e. from "North pole" to "South pole", try cutting it along the "Equator". Surprise! there's your Fibonacci number!
    3. Try a Sharon fruit.
    4. Where else can you find the Fibonacci numbers in fruit and vegetables? Why not email me with your results and the best ones will be put on the Web here (or linked to your own web page).

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Fibonacci Fingers?

Look at your own hand:

You have ...
  • 2 hands each of which has ...
  • 5 fingers, each of which has ...
  • 3 parts separated by ...
  • 2 knuckles

Is this just a coincidence or not?????

However, if you measure the lengths of the bones in your finger (best seen by slightly bending the finger)does it look as if the ratio of the longest bone in a finger to the middle bone isPhi?
What about the ratio of the middle bone to the shortest bone (at the end of thefinger) - Phi again?
Can you find any ratios in the lengths of the fingers thatlooks like Phi? ---or does it look as if it could be any other similar ratioalso?

Why not measure your friends' hands and gather some statistics?

NOTE: When this page was first created (back in 1996) this was meant as a joke and as something to investigate to show that Phi, a precise ratio of 1.6180339... is not "the Answer to Life The Universe and Everything" -- since we all know the answer to that is 42.
The idea of the lengths of finger parts being in phi ratios was posed in 1973 but two later articles investigating this both show this is false.
Although the Fibonacci numbers are mentioned in the title of an article in 2003, it is actually about the golden section ratios of bone lengths in the human hand, showing that in 100 hand x-rays only 1 in 12 could reasonably be supposed to have golden section bone-length ratios.
Research by two British doctors in 2002 looks at lengths of fingers from their rotation points in almost 200 hands and again fails to find to find phi (the actual ratios found were 1:1 or 1:1.3).
  • On the adaptability of man's hand J W Littler, The Hand vol 5 (1973) pages 187-191.
  • The Fibonacci Sequence: Relationship to the Human Hand Andrew E Park, John J Fernandez, Karl Schmedders and Mark S Cohen Journal of Hand Surgery vol 28 (2003) pages 157-160.
  • Radiographic assessment of the relative lengths of the bones of the fingers of the human hand by R. Hamilton and R. A. Dunsmuir Journal of Hand Surgery vol 27B (British and European Volume, 2002) pages 546-548

[with thanks to Gregory O'Grady of New Zealand for these references and the information in this note.]

Similarly, if you find the numbers 1, 2, 3 and 5 occurring somewhere it does not always means the Fibonacci numbers are there (although they could be).Richard Guy's excellent and readable article on how and why people draw wrong conclusions from inadequate data is well worth looking at:

  • The Strong Law of Small Numbers Richard K Guy in The American MathematicalMonthly, Vol 95, 1988, pages 697-712.

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Always Fibonacci?

But is it always the Fibonacci numbers that appear in plants?

I remember as a child looking in a field of clover for the elusive 4-leaved clover -- and finding one.
A fuchsia has 4 sepals and 4 petals:
and sometimes sweet peppers don't have 3 but 4 chambers inside:

and here are some flowers with 6 petals:


crocus

narcissus

amaryllis
You could argue that the 6 petals on the crocus, narcissus and amaryllis are really two sets of 3 petals if you look closely, and 3 is a Fibonacci number.However, the 4 petals of the fuchsia really shows there are plants with petals that are definitely not Fibonacci numbers.Four is particularly unusual as the number of petals in plants, with 3 and 5 definitely being much more common.

Here are some more examples of non-Fibonacci numbers:

Here is a succulent with a clear arrangement of 4 spirals
in one direction and 7 in the other:
and here is another with 11 and 18 spirals:whereas this Echinocactus Grusonii Inermis
has 29 ribs:

So it is clear that not all plants show the Fibonacci numbers!

Another common series of numbers in plants are theLucas Numbers that start off with 2 and 1 and then,just like the Fibonacci numbers, have the rule that the next is the sum of the two previous ones to give:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

Did you notice that 4, 7, 11, 18 and even 29 all occurred in the non-Fibonacci pictures above?

But, no matter what two numbers we begin with, the ratio of two successive numbers in all of theseFibonacci-type sequences always approaches a special value, the golden mean, of 1.6180339...and this seems to be the secret behind the series. There is more on this and how mathematics hasverified that packings based on this number are the most efficient on the next page at this site.

  • A sunflowerwith 47 and 76 spirals is an illustration from:
  • Quantitative Analysis of Sunflower Seed Packing by G W Ryan, J L Rouse and L A Bursill, J. Theor. Biol. 147 (1991) pages 303-328
  • Variation In The Number Of Ray- And Disc-FloretsIn Four Species Of CompositaeP P Majumder and A Chakravarti, Fibonacci Quarterly14 (1976) pages 97-100.
    In this article two students at the Indian Statistical Institute in Calcutta find that"there is a good deal of variation in the numbers of ray-florets and disc-florets" butthe modes(most commonly occurring values) are indeed Fibonacci numbers.

A quote from Coxeter on Phyllotaxis

H S M Coxeter, in his Introduction to Geometry (1961, Wiley, page 172) - see the references at the foot of this page - has thefollowing important quote:

it should be frankly admitted that in some plantsthe numbers do not belong to the sequence of f's [Fibonacci numbers]but to the sequence of g's [Lucas numbers] or even to the still moreanomalous sequences

3,1,4,5,9,... or 5,2,7,9,16,...

Thus we must face the fact that phyllotaxis is really not auniversal law but only a fascinatingly prevalenttendency.

But the tendency has behind it a universal number, the golden section,which we will explore on the next page.
  • He cites A H Church's The relation of phyllotaxis to mechanicallaws, Williams and Norgate, London, 1904, plates XXV and IXas examples of the Lucas numbers and platesV, VII, XIII and VI as examples of the Fibonacci numbers on sunflowers.

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References and Links

  • a book
  • an article, usually in an academic periodical
  • a link to a web page
Excellent books which cover similar material to that which you have found on this pageare produced by Trudi Garland and Mark Wahl:
  • Mathematical Mystery Tour by Mark Wahl, 1989, is full of many mathematicalinvestigations, illustrations, diagrams, tricks, facts, notes as wellas guides for teachers using the material. It is a great resource foryour own investigations.
  • Fascinating Fibonaccis by Trudi Hammel Garland.
    This is a really excellent book - suitable for all, and especially good for teachers seeking more material to use in class.

    Trudy is a teacher in California and has some more information on her book. (You can even Buy it online now!)
    She also has published several posters, including one on the golden section suitable for a classroom or your study room wall.
    You should also look at her other Fibonacci book too:

  • Fibonacci Fun: Fascinating Activities with Intriguing Numbers Trudi Hammel Garland - a book for teachers.
  • Mathematical Models H M Cundy and A P Rollett, (third edition, Tarquin, 1997) is still a good resource book though it talks mainly about physical models whereas today we might use computer-generated models. It was one of the first mathematics books I purchased and remains one I dip into still. It is an excellent resource on making 3-D models of polyhedra out of card, as well as on puzzles and how to construct a computer out of light bulbs and switches (no electronics!) which I gave me more of an insight into how a computer can "do maths" than anything else. There is a wonderful section on equations of pretty curves, some simple, some not so simple, that are a challenge to draw even if we do use spreadsheets to plot them now.
  • On Growth and Form by D'Arcy Wentworth Thompson, Dover, (Complete Revised edition 1992) 1116 pages. First published in 1917, this book inspired many people to look for mathematical forms in nature.
  • Sex ratio and sex allocation in sweat bees (Hymenoptera: Halictidae) D Yanega, in Journal of Kansas Entomology Society, volume 69 Supplement, 1966, pages 98-115.
    Because of the imbalance in the family tree of honeybees, the ratio of male honeybees to females is not 1-to-1. This was noticed by Doug Yanega of the Entomology Research Museum at the University of California. In the article above, he correctly deduced that the number of females to males in the honeybee community will be around the golden-ratio Phi = 1.618033..
  • On the Trail of the California Pine, Brother Alfred Brousseau, Fibonacci Quarterly, vol 6, 1968, pages 69 - 76;
    on the authors summer expedition to collect examples of all the pines in California and count the number of spirals in both directions, all of which were neighbouring Fibonacci numbers.
  • Why Fibonacci Sequence for Palm Leaf Spirals? in The Fibonacci Quarterlyvol 9 (1971), pages 227 - 244.
  • Fibonacci System in Aroids in The Fibonacci Quarterlyvol 9 (1971), pages 253 - 263. The Aroids are a family of plants that includethe Dieffenbachias, Monsteras and Philodendrons.
  • Phyllotaxis - An interactive site for the mathematical study of plant pattern formation by Pau Atela and Chris Golé of the Mathematics Dept at Smith College, Massachusetts.
    is an excellent site, beautifully designed with lots of pictures and buttons to push for an interactive learning experience! A must-see site!
  • Alan Turing
    one of the Fathers of modern computing (who lived here in Guildford during his early school years) was interested in many aspects of computers and Artificial Intelligence (AI) well before the electronic stored-program computer was developed enough to materialise some of his ideas. One of his interests (see his Collected Works) was Morphogenesis, the study of the growing shapes of animals and plants.
  • The book Alan Turing: The Enigma by Andrew Hodges is an enjoyable and readable account of his life and work on computing as well as his contributions to breaking the German war-time code that used a machine called "Enigma".
    Unfortunately this book is now out of print, but click on the book-title link and Amazon.com will see if they can find a copy for you with no obligation.
  • The most irrational number
    One of the American Maths Society (AMS) web site's What's New in Mathematics regular monthly columns. This one is on the Golden Section and Fibonacci Spirals in plants.
  • Phyllotaxis
    An interactive site for the mathematical study of plant pattern formation for university biology students at Smith College. Has a useful gallery of pictures showing the Fibonacci spirals in various plants.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

Navigating through this Fibonacci and Phi site

The Lucas numbers are formed in the same way as the Fibonaccinumbers - by adding the latest two to get the next, but instead ofstarting at 0 and 1 [Fibonacci numbers] the Lucas number series starts with 2 and 1. The other two sequences Coxeter mentions above have otherpairs of starting values but then proceed with the exactly the same rule as theFibonacci numbers. These series are theGeneral Fibonacci series.

An interesting fact is that for all series that are formed fromadding the latest two numbers to get the next starting from anytwo values (bigger than zero), the ratio of successive terms willalways tend to Phi!

So Phi (1.618...) and her identical-decimal sister phi (0.618...) are constants common to allvarieties of Fibonacci series and they have lots of interesting properties of their own too. The links above will take you to further pages on this site for you to explore. You can also just follow the links below in the Where To next?section at the bottom on each page and this will go through the pages in order.Or you can browse through the pages that take your interestfrom the complete collection and brief descriptions on the home page. There are pages onWho was Fibonacci?, the golden section (phi) inthe arts: architecture, music, pictures etcas well as two pages of puzzles.

Many of the topics we touch on in these pages open up new areas of mathematics such as ContinuedFractions, Egyptian fractions,Pythagorean triangles, and more, allwritten for school students and needing no more mathematics than is coveredin school up to age 16.
© 1996-2016 Dr Ron Knott
updated 25 September 2016

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