Fibonacci was born in the 13th century, in 1170 (approximately) and that he died in 1250.He was born in Italy but obtained his education in North Africa. Very little is known about him or his family and there are no photographs or drawings of him. Much of the information about Fibonacci has been gathered by his autobiographical notes which he included in his books.
However, Fibonacci is considered to be one of the most talented mathematicians for the Middle Ages. Few people realize that it was Fibonacci that gave us our decimal number system (Hindu-Arabic numbering system) which replaced the Roman Numeral system. When he was studying mathematics, he used the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have 0's and lacked place value. In fact, when using the Roman Numeral system, an abacus was usually required. There is no doubt that Fibonacci saw the superiority of using Hindu-Arabic system over the Roman Numerals. He shows how to use our
It was this problem that led Fibonacci to the introduction of the Fibonacci Numbers and the Fibonacci Sequence which is what he remains famous for to this day. The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This sequence, shows that each number is the sum of the two preceding numbers. It is a sequence that is seen and used in many different areas of mathematics and science. The sequence is an example of a recursive sequence. The Fibonacci Sequence defines the curvature of naturally occurring spirals, such as snail shells and even the pattern of seeds in flowering plants. The Fibonacci sequence was actually given the name by a French mathematician Edouard Lucas in the 1870's.
Contributions
Fibonacci is famous for his contributions to number theory.
Fibonacci is famous for his contributions to number theory.
· In his book, Liber abaci he introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.
· He introduced us to the bar we use in fractions, previous to this, the numerator has quotations around it.
· The square root notation is also a Fibonacci method.
It has been said that the Fibonacci numbers are Nature's numbering system and apply to the growth of living things, including cells, petals on a flower, wheat, honeycomb, pine cones and much more.
INTRODUCTION (FIBONACCI IN NATURE )
Ø The fibonacci numbers are Naturals’ numbering system.
Ø They appear everywhere in Nature,from the leaf arrangement in plants,to the patterns of florets of a flower,the bracts of a pinecone,or the scale of pineapple.
Ø The Fibonacci numbers are therefore applicable to the growth of every living thing,including a single cell,a grain of wheat,a hive of bees and even all of mankind.
Ø Plants do not know about this sequence
Ø They just grow in the most efficient ways with it own arrangement.
Ø Many plants show the Fibonacci Numbers in the arrangement of the leaves around the stem.
Ø Some pine cones and fir cones also show the numbers as do daisies and sunflowes
Ø Many other plants,such as succulents,also show the numbers.
Ø In the case of leaf arrangement, or (phyllotaxis), some of the cases may be related to maximizing the space for each leaf, or the average amount of light falling on each one. Even a tiny advantage would come to dominate, over many generations. In the case of close-packed leaves in cabbages and succulents the correct arrangement may be crucial for availability of space
.
Plants
· Plants illustrate the Fibonacci series in the numbers and arrangement of petals,leaves,section and seeds.
· Plants that are formed is spirals,such as pineapples,illustrate Fibonacci numbers.
· Referred to Mathematical Idea book ninth edition written by Miller, most pineapples exhibit the fibonacci sequence in the following way:Count the spirals formed by the ‘scales’ of the cones,first counting from lower left to upper right.
· (Phyllotaxis) is the study of the ordered position of leaves on a stem. The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities.
· We would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
· According to Mathematics For Elementary teachers,sixt edition it state that the seed for example in in the centre of daisy are arranged in two intersecting sets of spiral,one turning clockwise and the other turning anticlockwise.the number of spiral in each sets is Fibonacci number.Also the number of petals wili often be a Fibonacci number.
· We can conclude Fibonacci state that:
Plants illustrate the series in the number and arrangement of
ü Petals
ü Leaves/branches
ü Section
ü Seed
FIBONACCI NUMBERS IN PLANT
[BRANCHING/LEAVES AND SPIRALS]
Fibonacci numbers in plant branching Fibonacci numbers in plant spirals
| Here are plant illustrates that each successive level of branches is often based on a through the fibonacci series. |
| Here are pineapple illustrates in plant spirals because they are roughly hexagonal in shape, three distinct sets of spirals may be observed. |
| Fibonacci Number in plant leaves | |
| |
| The number of leaves in any stage will also be a Fibonacci number. |
We might expect symmetry in plants, but if we cut a fruit or vegetable we will often we might expect symmetry in plants.
But if we cut a fruit or vegetable we will often find that the number of sections is a Fibonacci number:
Example:
Banana have 3 Apple have 5
[ PETAL ON FLOWER]
Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number:
We can understand it based on the example given:
| 3 | Lilies | |
| 5 | Buttercups, Roses | |
| 8 | Delphinium | |
| 13 | Marigolds | |
| 21 | Black-eyed susans | |
| 34 | Pyrethrum | |
| 55/89 | Daisies |
Based on wikipedia,in mathematics,the Fibonacci numbers in the following integer sequences:
0,1,1,2,3,5,8,13,21,34,55,89,144,...this sequences of Fibonacci numbers arises all over mathematics and also in nature.
By the definition,the first two Fibonacci numbers are 0 and 1,and each subsequent number is the sum of the previous two.Some sources omit the initial 0,instead beginning the sequence with two 1s.
In mathematical terms,the sequence Fn of Fibonacci numbers is defined by the recurrence relation
Fn= Fn-1 + Fn-2
With seed values
F0 = 0 and F1=1
However,if we wanted the 100th terms of this sequence,it would take lots of intermediate calculations with the recursive formula to get a result.
We may use the n-th term,that is
| an = [Phin – (phi)n ]/sqrt[5] |
where Phi= (1+sqrt [5])/2 is the so called golden mean
while phi= (1-sqrt[5])/2 is an associated golden number,also equal to (-1/phi).
This formula is attributed to Binet in 1843,though known by Euler before him.
A more abstract way of putting it is that the Fibonacci numbers fn are given by the formula f1 = 1, f2 = 2, f3 = 3, f4 = 5 and generally f n+2 = fn+1 + fn . For a long time, it had been noticed that these numbers were important in nature, but only relatively recently that one understands why. It is a question of efficiency during the growth process of plants.
PATTERNS IN NATURE
According to Mathematics for elementary teachers,sixth edition:
The spiral is a common pattern in nature.It can found in plant,animal and others.
The frequent occurrence of spirals in living things can be explained by different growth rates.
Living form curl because the faster growing [longer] surface lies outside and the slower growing [shorter] surface lies inside.
A variety of patterns occur in plants and trees.
Many of these patterns are related to a famous sequence of numbers called the Fibonacci Number.
After the two numbers of this sequence,which are 1and 1,each successive number may be obtained by adding the two previous numbers.
1,1,2,3,5,8,13,21,34,55,....
The seeds in the center of a daisy are arranged in two intersecting sets of spirals,one turning clockwise and the other turning counterclockwise.
The number of spirals in each sets is a Fibonacci number.Also the number of petals wili often be a Fibonacci number.
shasta daisy with 21 petals
SOME OF APPLICATION FROM FIBONACCI ABOUT NUMBERS OF PLANTS
Leaf arrangements
· 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 exampleØ In the top of plant , 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.
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.
Ø 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).
For more detail.. (look at the diagram given)
| The schematic diagrams of the sneezewort and tree have been presented as though the plants were flat. This illustrates the development which leads to Fibonacci numbers, but it suppresses the characteristic of a majority of plants that successive leaves or shoots spiral around the main stem as successive stages develop. Suppose we fix our attention on some leaf on the bottom of a stem on which there is a single leaf at any one point. | | ||
| | |||
| If we number that leaf "0" ... | |||
| | |||
| And count the leaves up the stem until we come to the one which is directly above the starting one, the number we get is generally a term of the Fibonacci sequence. | |||
| Again as we work up the stem, let us count the number of times we revolve about it. | | ||
| | |||
| | |||
| This number, too, is generally a term of the sequence. | | ||
| The leaves here are numbered in turn, each exactly 0.618 of a clockwise turn (222.5°) from the previous one. | |||
| Leaf number | turns clockwise |
| 3 | 1 |
| 5 | 2 |
| 8 | 3 |
We 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 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
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.
There are ( Pineapple).what are related between pineapple and Fibonacci number of plants?| Pineapple scales are also patterned into spirals . | |
| because they are roughly hexagonal in shape, three distinct sets of spirals may be observed. | |
| One set of 5 spirals ascends at a shallow angle to the right, ... | |
| a second set of 8 spirals ascends more steeply to the left, ... | |
| and the third set of 13 spirals ascends very steeply to the right. |
| |
Quite amazing that the Fibonacci number patterns occur so frequently in nature
( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature". Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of Fibonacci numbers or golden sections in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves The Fibonacci numbers are also found in the family tree of honeybees..
Conclusion:
v We can conclude that Fibonacci number of plants is:
Plants illustrate the series in the number and arrangement of
ü Petals
ü Leaves/branches
ü Section
ü Seed
There are Some of information about Navigating through 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 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.
v We know that Fibonacci nature’ system was apply to the growth of living things, including cells, petals on a flower, wheat, honeycomb, pine cones and much more..
v Based on the Fibonacci numbers of plants,actually we should be open – mind and see the truth and powerfull of our God by his creater and try to inverstigate all of the living things in this world especially about the nature that was found by Fibonacci.
v We also can applicate it not only in plants but even all of our God creaters.
v Actually,this world have many of secret that ask us to find and investigate all of living things that have in it.
v Fibonacci should be a part of our model as he is mathematician that share with us about the numbers of plants and others.So,we ought to follow by his step to open the secret all nature in this world.
SELF REFLECTION
Alhamdulillah,praise to be Allah because finally I had Finished this coursework successfully according to the dateline that had given.I am also grateful because finally I had done this task completely follow the submission date that had given.Thank you to my mathematics’lecturer,Puan Fazidah because gave us an extra time to finish our this mathematics coursework to make it be more completely.
According to the dateline that had given,I had try the best to do this assignment be more complete and successfully ,but there are some of problem that I had face which disturb me in a process to finished this coursework.Honestly, that is not only trouble that I had face, I also got some of benefits by doing this assignment Actually,this coursework gave me more of benefits compare to a problem.
Based on our topic of numerical literacy that is about “Fibonacci Numbers of Plants”,I got many of information and knowledge about this topic.Honestly,I never know about Fibonacci and his concept before it,but the task that had given gave me knowledge all about him and give me a chance to gain my knowledge about mathematics history.His concept actually help me to understand and try to aplicate in my life about all living thing in this world.Beside that,I also got some of information about mathematics history not only about him but all the mathematician.In a process to find the sources that related to my topic,I took a chance to read some of information about other mathematics history and it gave me more knowledge.In addition,This coursework teach me to be more discipline with a time because I must submit this task follow the dateline that had given.So, I try to obey a time and determination do this coursework.Finally,This assignment help me to improve my grammar and than help me to aplicate it in others mathematics task.
That is difficult for me to find the souces of our topics that had given because I must find the different sources as a reference.It is some of problem that I had face among do this task.Actually our library have not much of book about history of mathematics and it give me a trouble and I try to find other sources from many alternative like internet .Beside that,The topic that I got actually very difficult to me to understand . It give me a problem because how I want to do this task if I do no what its about? What’s it concept? And all about the related about Fibonacci Numbers of Plants.So, I try to ask my friend all about Fibonacci concept to give me little information and I was started this task after undestand about his concept.Finally,this coursework given actually clash with other coursework that I must to complete and submit in same date.So,It give me a problem and a challengers to me to do all the tasks according the dateline that had given.Because of it,I have no much time to complete all assignment and I try the best to finish it completely.
According to all a problem that I had faced,I have some of ways that used to overcome it to make it complete follow the submission date.First,I always obey a time that I make it by myself .Second I always be determination and responsible to finish this task completely follow the submission date and finally I always be discipline by all the work that I had done..
REFERENCE
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