Fibonacci Sequence in Nature free essay sample

Fibonaccis 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 dieand 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? 1. At the end of the first month, they mate, but there is still one only 1 pair. 2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. 3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. 4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, Another view of the Rabbits 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 parents 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. Dudeneys 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 Fibonaccis 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 lets look at another real-life situation that is exactly modelled by Fibonaccis series honeybees. 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 honeybees 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 queens 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. Lets look at the family tree of a male drone bee. 1. He had  1  parent, a female. 2. He has  2  grand-parents, since his mother had two parents, a male and a female. 3. He has  3  great-grand-parents: his grand-mother had two parents but his grand-father had only one. 4. How many great-great-grand parents did he have? Again we see the  Fibonacci numbers  : great- great,great gt,gt,gt grand- grand- grand 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 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 an be found with 34, 55 or even 89 petals. The links here are to various flower and plant catalogues: * the Dutch  Flowerwebs searchable index called  Flowerbase. * The US Department of Agricultures  Plants Database  containing over 1000 images, plant information and searchable database. Fuchsia| Pinks| Lily| 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. Marks  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)     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  The humble buttercup has been bred into a multi-petalled form. 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. Seed heads This po ppy 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 Stones 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 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. | 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! 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. 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. 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 next page. | 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!