Nature Of Mathematics 13th Edition Answer Key
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 
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 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:
- 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.
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 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 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!
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!
Males are produced by the queen's unfertilised eggs, so male bees only have a mother but no father!
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.
- 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. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
You do the maths...
- 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!
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
Fibonacci numbers and the Golden Ratio
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:
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] .
You do the maths...
- 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!
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
. 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 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.
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-circles; 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 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.
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.
- The Curves of Life Theodore A Cook, Dover books, 1979, ISBN 0 486 23701 X.
A Dover reprint of a classic 1914 book.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
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.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
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; 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.
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 - 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:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
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.
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:
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 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.
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. 
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.
You do the maths...
- 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: - 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?
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
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. |
You do the maths...
- 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?
- From St. Mary's College (Maryland USA), Professor Susan Goldstine
has a page with really good pine cone 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 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.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
Leaf arrangements
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. 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 |
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 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): |
You do the maths...
- 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 pine cone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the Fibonacci numbers.
- First look at it:
- 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 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 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 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.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:
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
- 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:But the tendency has behind it a universal number, the golden section,which we will explore on the next page.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 .
- 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.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..
References and Links
- a book
- an article, usually in an academic periodical
- a link to a web page
- 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.
- 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 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.
- 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 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 identical-decimal 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.
Many of the topics we touch on in these pages open up new areas of mathematics such as Continued Fractions, Egyptian fractions, Pythagorean triangles, and more, all written for school students and needing no more mathematics than is covered in school up to age 16.
© 1996-2016 Dr Ron Knott 
updated 25 September 2016
Nature Of Mathematics 13th Edition Answer Key
Source: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
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