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

The icon means there is a Things to do
investigation at the end of the section.

Rabbits,
Cows and Bees Family Trees

Fibonacci's
Rabbits..

Dudeney's
Cows

Honeybees
and Family Trees

Fibonacci
Numbers and the Golden Number

Fibonacci
Rectangles and Shell Spirals

Fibonacci
numbers, the Golden Section and plants

Petals
on flowers

Seed
heads

Pine
cones

Leaf
arrangements

Leaves
per turn

Leaf
arrangements of some common plants

Vegetables
and Fruit

Fibonacci
Fingers?

Always
Fibonacci?

A
quote from Coxeter on Phyllotaxis

Navigating
through this site

References
and Links

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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 the
answer!

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

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

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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 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 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

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 [uzzles, an excellent chapter on all
kinds of mazes and solving them, magic squares, river crossing puzzles, and
more, all with full soutions 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 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 ancestors (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 unfertilized 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 Fibonacci
numbers :

great- great,great gt,gt,gt
grand- grand- grand grand
Number of parents: parents: parents: parents: parents:
of a MALE bee: 1 2 3 5 8
of a FEMALE bee: 2 3 5 8 13

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 year 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
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?

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 a later page [this link opens a new window] .

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, ..?

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 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 units long; 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 arrangment 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 center 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 boyancy in
the water. Click on the picture to enlarge it in a new window. Draw a line from
the center 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.

Here are some more posters available from AllPosters.com 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
Wampler, Sondra
Buy this Art Print at AllPosters.com

Nautilus
Shell
Myers, Bert
Buy this Art Print at AllPosters.com

Nautilus
Schenck, Deborah
Buy this Art Print at AllPosters.com

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

Reference

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 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 petals; lilies and iris have 3 petals; some delphiniums have 8; corn
marigolds have 13 petals; 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:

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.

Fuchsia

Pinks

Lily

Daisies
available as a
poster at
AllPosters.com

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 petals they have - eg 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
Echinacea purpura and
native to the Illinois prairie where he lives.

You can have a look at some more of Tim's wonderful photographs on the web.

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. 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:

Sunflower
Buy This Art Print At
AllPosters.com

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.

Sunflower
Buy This Poster At
AllPosters.com

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 500, 1000 and 5000 seeds.

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.]

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 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 : An
Interactive Site for the Mathematical Study of Plant Pattern Foramtion 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: Nicky's Seeds
who supplies the whole range of flower and vegetable seed including sunflower
seed in the UK.

Have a look at the online catalogue at Nicky's Seeds 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 pinecone 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?

Use these buttons to check your answer (the lines are drawn connecting the
centres of each segment of the pinecone):

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 pinecone.

Show only the pinecone

Show the segment edges

Show the outline only

Show one set of spiral

Show the other set of spirals

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 Susan Goldstine

has a page with really good pinecone
pictures 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 soetimes 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.

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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.
Here's a
computer-generated image, based on an African violet type of plant, whereas
this
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 (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 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?

Leaf
number

turns
clockwise

3

1

5

2

8

3

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:

1/2 elm, linden, lime, grasses
1/3 beech,
hazel, grasses, blackberry
2/5 oak, cherry, apple, holly, plum,
common groundsel
3/8 poplar, rose, pear, willow
5/13 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 Home
page has links to lots of pictures.

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

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 organized 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 Brocolli/Cauliflower (or Romanesco) looks and tastes like a cross
between brocolli 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 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?

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 pinecone 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), ie from "North pole" to "South pole",
try cutting it along the "Equator". Surprise! there's your Fibonacci number!

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).

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

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 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?

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
artcle 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 occuring 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 conculsions 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.

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

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 unusal 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 Lucas
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 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 the
next page at this site.

A sunflower
with 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

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.

Navigating through this Fibonacci and Phi site

The Lucas
numbers 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 General
Fibonacci series.

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 sister phi (0.618...) are constants common to all
varieties 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
interest from the complete collection and brief descriptions on the home
page. There are pages on Who
was Fibonacci?, the golden section (phi) in the
arts: architecture, music, pictures etc as well as two pages of puzzles.

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

References and Links

Key
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 to 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:

Mathematical
Mystery Tour 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:

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 oneof 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 outof light bulbs ans 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 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

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 materialize 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.

There are no earlier topics - this is the first.

the
Fibonacci Home Page

This is the first page on this Topic.

Where to now?

The next page on this topic is ...
The
golden section in nature

The next Topic is...
The
Puzzling World of Fibonacci Numbers

©
1996-2007 Dr Ron
Knott updated 23 June 2007