A man named Leonardo Pisano, who was known by his nickname, "Fibonacci", and named the series after himself, first discovered the Fibonacci sequence around 1200 A.D. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in Italy, but the exact dates of his birth and death are not known. He played an important role in reviving ancient mathematics and made significant contributions of his own. Even though he was born in Italy, he was educated in North Africa where his father held a diplomatic post. He published a book called Liber abaci, in 1202, after his return to Italy and it was in this book that the Fibonacci numbers were first discussed. It was based on bits of Arithmetic and Algebra that Fibonacci had accumulated during his travels with his father. Liber abaci introduced the HinduArabic placevalued decimal system and the use of Arabic numerals into Europe. Though people were interested, this book was somewhat controversial because it contradicted some of the foremost Roman and Grecian Mathematicians of the time, and even proved many of their calculations to be false.
The Fibonacci sequence is also used in the Pascal triangle. The sum of each diagonal row is a Fibonacci number. They are also in the right sequence.
The Fibonacci sequence has been a big factor in many patterns of things in nature, which is quite fascinating. It's been discovered that the numbers representing the screwlike arrangements of leaves on flowers and trees are very often numbers in the Fibonacci sequence. On many plants, for instance, 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...
...FibonacciSequenceFibonacci, also known as the Leonardo of Pisa, born in the early 1770’s AD in Pisa, Italy, has had a huge impact on today’s math, and is used in everyday jobs all over the world. After living with his dad, a North African educator, he discovered these ways of math by traveling along the Mediterranean Coast learning their ways of math. With the inspiration from the “HinduArabic” numerical system, Fibonacci created the 09 number system we still use to this day.
One of his most important and interesting discoveries is probably what is known as the Fibonaccisequence. It goes like this: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. He discovered this sequence through an experiment on an over population and breeding of rabbits. He then realized that if you add the last two numbers together you get the next one.
The Fibonaccisequence can be found almost anywhere including: architecture, economics, music, aesthetics, and most famously known, nature. For example the way seeds are arranged on a sunflower or pinecone, uses the Fibonaccisequence to prevent over crowding. It can also be seen in spiral galaxies, shells, the way water falls on a spider web, and even in your own body. Did you know that if you go from the bone in the tip of your finger to it’s middle it should be two...
...patterns in nature focusing in the Fibonaccisequence as a main and looking for angles. What was first done was to count a pine cone’s pieces, a flower’s petals, a celery, and grapes to find the Fibbonacci sequence which not found only on the celey and on the flower, elsewhere the Fibonacci was there.
After finishing the experiment I started noticing more patterns relating to the Fibonaccisequence. For example, in a tree you start counting by the tree trunk; if you start going up there are two branches with three leaves, then five, them eight until there is no more to count you go to the next branch and do the same thing until you reach the top of the tree. I think math can be found practically everywhere you look if you can find the right sequence. When you are looking for patterns there is at least one for anything. Math can be very important and people can start caring more about it if they know it is all around them.
Introduction
In my science fair project I am going to try to find mathematical patterns in nature. The main pattern I am looking for is for the Fibonaccisequence, which consist of the numbers in the following order: 1,1,2,3,5,8,13,21,34… so forth and so on always adding the number before. I will try my experiments in trees, pine combs, flowers, fruits, seashells, and vegetables. I think that the Fibonacci...
...Anatolia College 
Mathematics HL investigation

The Fibonaccisequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonaccisequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonaccisequence
The Fibonaccisequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value...
...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 newlyborn 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 Rabbit's Family Tree:
  
Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows:
* All the rabbits born in the same month are of the same generation and are...
...
The Fibonaccisequence
The Fibonaccisequence is a series of numbers developed by Leonardo Fibonacci as a means of solving a practical problem. 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 question 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 Fibonaccisequence is the series of numbers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… When squares are made with the widths, you get a nice spiral. If you look closely at the...
...
It can be clearly seen from the graph that the pattern/structure is exponential. This is due to the previous numbers being added in succession with the next, resulting in the ‘gap’ between each number to increase.
The trend in which the numbers follow is called a Fibonaccisequence and is often found in nature as well.
Many instances in which the Fibonacci Series is present in nature are that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. However some plants such as the sneezewort plant (as seen left) can be seen demonstrating the Fibonacci pattern in succession. It happens on both the number of stems and number of leaves.
Another appearance of the Fibonacci Series in nature is that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. This includes the pineapple shown to the left. The number of spirals going in each direction is a Fibonacci number. For example, there are 13 spirals that turn clockwise and 21 curving counter clockwise. On all other sunflowers, the number of clockwise and counter clockwise spirals will always be consecutive Fibonacci Numbers like 21 and 34 or 55 and 34.
Due to the rabbits’ problem not being very realistic, there are some concerns about the accurateness...
...SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
* For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context.
ARITHMETIC SEQUENCE
* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term...
...many years, particularly that of the Fibonaccisequence and the Golden Ratio. In Debussy’s Nocturne, composed in 1892, I look into the use of the Fibonaccisequence and the Golden Ratio. Previously it has been noted that composers used the Fibonaccisequence and the Golden Ratio in terms of form, however in my analysis I look into the use of it in terms of notation as well. I will explore how the idea of Sonata form is used along with the Mathematical Model of the Fibonaccisequence. It is however important to mention that as this is one of Debussy’s earlier works, the extent that the ratio and sequence are explored are not as elaborate as some of his later works. I will explore the Harmonic analysis of the piece to create a better understanding of where and how structure is used by Debussy. Debussy was a perfectionist and would only give perfected scores to the printers, as such it is impossible to prove whether or not the use of the sequences were intended or not, however considering that some of his contemporaries in other arts were very much involved with the idea of the Golden Ratio it does seem plausible that it was intended. The fact remains, though, that the use of the sequences and ratio are still evidently there and can be analysed; as this essay will show.
The Fibonaccisequence is...
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