Fibonacci Numbers and Nature
This page has been split into TWO PARTS.
This, the first, looks at the Fibonacci
numbers and why they appear in various "family trees" and patterns of spirals
of 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
icon means
there is a
Things to do investigation at the end of the section.
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Rabbits, Cows and Bees Family Trees
Let's look first at the Rabbit Puzzle that Fibonacci wrote about and then at two
adaptations of it to make it more realistic.
This introduces you to the Fibonacci Number
series and the simple definition of the whole never-ending series.
Fibonacci's Rabbits
The original problem that Fibonacci investigated (in the year
1202) was about how fast rabbits could breed in ideal circumstances.
Suppose a newly-born pair of rabbits, one male, one female, are
put in a field. Rabbits are able to mate at the age of one month so that
at the end of its second month a female can produce another pair of
rabbits. Suppose that our rabbits never die and that
the 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?
At the end of the first month, they mate, but there is still
one only 1 pair.
At the end of the second month the female produces a new pair,
so now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a
second pair, making 3 pairs in all in the field.
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 !
are here and some questions for you to answer.
Now can you see why this is the answer to our
Rabbits problem? If not,
Another view of the Rabbit's Family Tree:
Both diagrams above represent the same information.
Rabbits have been numbered to enable
comparisons 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 pages
at this site.
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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 that
the female of each pair mates with any male and produces another
pair.
Another problem which again is not true to life, is that each birth is of
exactly 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's
Rabbits 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 into
years and rabbits into bulls (male) and cows (females) in problem 175 in his book
536 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
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 bearing
his name does have lots of other interesting and practical
applications 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
, Dover Press,
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!
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.
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 live
solitary 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 show
how the
Fibonacci numbers count a honeybee's
(in this section a "bee" will mean a "honeybee").
First, some
unusual facts about honeybees such as: not all of them have two
parents!
In a
colony of honeybees there is one special female called the
queen.
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 so
have two parents. Females usually end up as worker bees but some are
fed with a special substance called royal jelly which makes
them grow into queens ready to go off to start a new colony when the
bees form a swarm and leave their home (a
hive) in search of a place to build a new nest.
So female bees have 2 parents, a male and a female whereas male
bees have just one parent, a female.
Here we follow the convention of Family Trees
that parents appear above their children, so the latest generations are
at the bottom and the higher up we go, the older people are.
Such trees
show all the ancestors (predecessors, forebears, antecedents) of the person
at the bottom of the diagram.
We would get quite a different tree if we listed all the descendants
(progeny, offspring) of a person
as 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.
He had 1 parent, a female.
He has 2 grand-parents, since his mother had
two parents, a male and a female.
He has 3 great-grand-parents: his
grand-mother had two parents but his grand-father had only one.
How many great-great-grand parents did he have?
Again we see the
great,great
grand
Number of
parents:
of a MALE bee:
8
of a FEMALE bee:
The Fibonacci Sequence as it appears in
Nature by S.L.Basin in Fibonacci Quarterly, vol 1 (1963), pages 53 - 57.
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Things to do
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
and place of birth and death
and also the dates of any marriages.
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.]
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?
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
..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?
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 Number
If we take the ratio of two successive numbers in Fibonacci's
series, (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 a
graph:
The ratio seems to be settling down to a particular value, which
we call the golden ratio or the golden number.
It has a value of approximately
1·618034 ,
although we shall find an even more accurate value on
Things to do
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, ..?
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 numbers
1,1,2,3,5,8,13,21,.. if we start with two small squares of size 1
next 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 the
latest square of side 2 - so having sides 3 and then
another touching both the 2-square and the 3-square (which has sides
of 5 units). We can continue adding squares around the picture,
each new square having a side which is as long as the sum of the
latest two square's sides. This set of rectangles whose sides are
two successive Fibonacci numbers in length
and which are composed of squares with sides which are
Fibonacci numbers, we will call the
Fibonacci Rectangles.
Here is a spiral drawn in the squares, a quarter of a circle in each square.
The spiral is
not a true
mathematical spiral (since it is made up of fragments which are parts of circles and does not go on
getting 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 of
the golden number in each square.
So points on the spiral are 1.618 times as far from the centre
after a quarter-turn.
In a whole turn the points on a radius out from the centre
are 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-heads
referring to D'Arcy Thompson's On Growth and Form probably meaning chapter 6
"The Equiangular Spiral".
Here Thompson
is
talking about a class of spiral with a constant expansion factor along a central line and not just shells with a Phi
expansion 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 provide
buoyancy 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 two
places where the shell crosses it so that the shell spiral has gone round just once
between 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 not
being 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 approximation
as it is made up of separate and distinct
quarter-
secondly the (true) spiral increases by a factor Phi every quarter-turn
so 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 posters
available
that are great for your study wall or classroom
or to go with a science project.
Click on the pictures to enlarge them in a new window.
Nautilus Shell
The curve of this shell is called
and are common in nature, though the 'growth factor' may not always be the golden ratio.
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
One plant in particular shows the Fibonacci numbers in the number
of "growing points" that it has. Suppose that when a plant puts out a
new shoot, that shoot has to grow two months before it is strong
enough to support branching. If it branches every month after that at
the 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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Petals on flowers
On many plants, the number of petals is a Fibonacci number:
buttercups have 5 lilies and iris have 3 some
delphiniums have 8; corn marigolds have 13 some asters have
21 whereas daisies can be found with 34, 55 or even 89 petals.
The links here are to various flower and plant catalogues:
's searchable index called
The US Department of Agriculture's
containing over 1000 images, plant information and
searchable database.
Pinks
Lily
Daisies
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.
contains many
flower pictures where the difference
between sepals and petals is clearly visible.
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.
buttercup, wild rose,
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,
55, 89 petals: michaelmas daisies,
the asteraceae family.
Some species are very precise about the number of petals
they have - e.g. buttercups, but others have petals that are very near
those 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 of
seeds 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 flower
in the picture
is about 2 cm across.
It is a member of the daisy family with the scientific name
and native to the Illinois prairie where he lives.
You can have a look
at some more of
Tim's wonderful
.
&
You can see that the orange "petals" seem to form spirals curving both to
the 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 shown
and the closest 3 seeds and a larger seedhead of 3000 seeds with the nearest seeds
shown. Each
clearly reveals the Fibonacci spirals:
A larger image appears in the book
Dara O' Briain (Author), 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.
Here is a sunflower with the same arrangement:
This is a larger sunflower with 89 and 55 spirals at the edge:
Here are some more wonderful pictures from All Posters (which you can buy
for your classroom or wall at home).
Click on each to enlarge it in a new window.
The same happens in many seed and flower heads in nature. The reason seems to
be 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 not
too sparse at the edges.
The spirals are patterns that the eye sees,
"curvier" spirals appearing near the centre, flatter spirals (and
more of them) appearing the farther out we go.
So the number of spirals we see, in either direction, is different for
larger flower heads than for small. On
a large
flower 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
Click on the
image on the right for a Quicktime animation of 120 seeds appearing
from 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 big
the seed head gets, the seeds are always equally spaced. At all
stages the Fibonacci Spirals can be seen.
The same pattern shown by these dots (seeds) is followed if the
dots then develop into leaves or branches or petals. Each dot
only 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 the
growing tip - the seeds only get bigger once they have appeared.
[This animation was produced by Maple. If there are N seeds in one
frame, then the newest seed appears nearest the central dot, at 0·618
of a turn from the angle at which the last appeared. A seed which is
i 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.]
by Roger V. Jean (400 pages, Cambridge University Press, 1994) has a good illustration
on its cover - click on the book's title link or this little picture of the cover
and on the page that opens,
click on picture of the front cover
to see it. It clearly shows that the spirals the eye sees are different near the centre
on a real sunflower
seed head, with all the seeds the same size.
Smith College (Northampton, Massachusetts, USA)
has an excellent website :
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 in
the number of petals or spirals on seed heads etc., although
they
often come close to the Fibonacci numbers.
Things to do
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:
who supplies the whole range of flower and vegetable seed including sunflower seed in the UK.
Have a look at the online catalogue at
where there are lots of
pictures of each of the flowers.
Which plants show Fibonacci spirals on their flowers?
Can you find an example of flowers with 5, 8, 13 or 21 petals?
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 picture
of 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.
Things to do
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?
What about a pineapple?
Can you spot the same spiral pattern?
How many spirals are there in each direction?
Links and References
From St. Mary's College (Maryland USA),
Professor
has a page with really
good
showing 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 of
spirals in one or both directions.
Sometimes this is due to deformities produced
by 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 of
different kinds.
The author also found that there were as many
with 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 Quarterly
vol 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 and
all exhibited 5,8 or 13 spirals.
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Leaf arrangements
Also, many plants show the Fibonacci numbers in the
arrangements of the leaves around their stems. If we look down on a
plant, the leaves are often arranged so that leaves above do not hide
leaves below. This means that each gets a good share of the sunlight
and catches the most rain to channel down to the roots as it runs
down the leaf to the stem.
based on an African violet type of plant, whereas
has lots of leaves.
Leaves per turn
The Fibonacci numbers occur
when counting both the number of times we go around the stem, going
from leaf to leaf, as well as counting the leaves we meet until we
encounter a leaf directly above the starting one.
If we count in the
other direction, we get a different number of turns for the same
number of leaves.
The number of turns in each direction and the
number of leaves met are three consecutive Fibonacci
numbers!
For example, in the top plant in the picture above, we have
3 clockwise rotations before we meet a leaf directly
above the first, passing 5 leaves on the way. If we
go anti-clockwise, we need only 2 turns. Notice that
2, 3 and 5 are consecutive Fibonacci numbers.
For the lower plant in the picture, we have 5
clockwise rotations passing 8 leaves, or just
3 rotations in the anti-clockwise direction.
This
time 3, 5 and 8 are consecutive numbers in the Fibonacci sequence.
We can write this as, for the top plant, 3/5 clockwise
rotations per leaf ( or 2/5 for the anticlockwise
direction). For the second plant it is 5/8 of a turn per leaf
The sunflower here when viewed from the top shows the same pattern. It is the same plant whose side view is above.
Starting at the
leaf 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?
Leafnumberturnsclockwise
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 all
plants exhibit this pattern of leaves involving the Fibonacci
numbers.
Some common trees with their Fibonacci leaf arrangement numbers
are:
elm, linden, lime, grasses
beech, hazel, grasses, blackberry
oak, cherry, apple, holly, plum, common
poplar, rose, pear, willow
pussy willow, almond
where t/n means each leaf is t/n of a turn after the last leaf or that there is
there 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 that
the Fibonacci numbers occur in Bromeliads and his
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 Fibonacci
numbers for yourself in vegetables and fruit.
Things to do
Take a look at a
cauliflower next time
you're preparing one:
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
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?
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
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.
Try the same thing for broccoli.
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.
Look for the Fibonacci numbers in fruit.
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.
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
Try a Sharon fruit.
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 ...
hands each of which has ...
fingers, each of which has ...
parts separated by ...
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 is
Phi?
What about the ratio of the middle bone to the shortest bone (at the end of the
finger) - Phi again?
Can you find any ratios in the lengths of the fingers that
looks like Phi?
---or does it look as if it could be any other similar ratio
also?
Why not measure your friends' hands and gather some statistics?
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
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.
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 Mathematical
Monthly, Vol 95, 1988, pages 697-712.
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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:
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 the
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,
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 these
Fibonacci-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 has
verified that
packings based on this number are the most efficient on
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-Florets
In Four Species Of Compositae
P P Majumder and A Chakravarti, Fibonacci Quarterly
14 (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"
but
the 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 the
following important quote:
it should be frankly admitted that in some plants
the 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 more
anomalous sequences
3,1,4,5,9,... or 5,2,7,9,16,...
Thus we must face the fact that phyllotaxis is really not a
universal law but only a fascinatingly prevalent
tendency.
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 mechanical
laws, Williams and Norgate, London, 1904, plates XXV and IX
as examples of the Lucas numbers
and plates
V, VII, XIII and VI as examples of the Fibonacci numbers on sunflowers.
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References and Links
means the reference is to a book (and any
link will take you to more information about the book and an on-line site from which you can
purchase it);
means the reference is to an
article in a magazine or a paper in a scientific periodical.
indicates a link
another web site.
Excellent books which cover similar material to that which you have found on this page
are produced by Trudi Garland and Mark Wahl:
by Mark Wahl, 1989, is full of many mathematical
investigations, illustrations, diagrams, tricks, facts, notes as well
as guides for teachers using the material.
It is a great resource for
your own investigations.
Books by Trudi Garland:
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
(You can even )
She also has published
including one on
suitable for
a classroom or your study room wall.
You should also look at her other Fibonacci book too:
Trudi Hammel Garland - a
book for teachers.
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
It was one of the first mathematics books I purchased and remains one I dip into still.
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.
by D'Arcy Wentworth Thompson, Dover,
(Complete Revised edition
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)
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 Quarterly
vol 9 (1971), pages 227 - 244.
Fibonacci System in Aroids in The Fibonacci Quarterly
vol 9 (1971), pages 253 - 263.
The Aroids are a family of plants that include
the Dieffenbachias, Monsteras and Philodendrons.
WWW links on Phyllotaxis, the Fibonacci
Numbers and Nature
Pau Atela and Chris Gol& 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!
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
was , the study of the growing shapes of animals and
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
will see if they can find a copy for you
with no obligation.
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.
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.
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Navigating through this Fibonacci and Phi site
are formed in the same way as the Fibonacci
numbers - by adding the latest two to get the next, but instead of
starting at 0 and 1 [Fibonacci numbers] the Lucas number series starts with 2 and 1.
The other two sequences Coxeter mentions above have other
pairs of starting values but then proceed with the exactly the same rule as the
Fibonacci numbers.
These series are the
.
An interesting fact is that for all series that are formed from
adding the latest two numbers to get the next starting from any
two values (bigger than zero), the ratio of successive terms will
always tend to Phi!
So Phi (1.618...) and her identical-decimal sister phi (0.618...) are constants common to all
varieties of
Fibonacci series and they have lots of
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 interest
from the complete collection and brief descriptions
There are pages on
, the golden section (phi) in
as well as two pages of .
Many of the topics we touch on in these pages
open up new areas of mathematics such as , ,
, and more, all
written for school students and needing no more mathematics than is covered
in school up to age 16.
updated 30 October 2010
